Who is popularly known as Kerala Vyasan ?

C) Kodungalloor Kunjikkuttan Thampuran

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100 questions with answersProvisional keyAssistant Grade

Statistical Assistant GR- II - Economics and Statistics question paper 2015

Date of test: 29-06-2015
Question paper code: 11/2015/OL
Category No.: 195/2013
Questions: 100
Answer key: Provisional Answer Key
Exam mode: Online (computer based)

Provisional key. Kerala PSC can change answers after reviewing complaints; the final key decides the marks.

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Source: Kerala PSC (keralapsc.gov.in) · official document· Last updated 25 Sept 2026· How we check content

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All 100 questions with answers

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1.
Drafting Committee chairman of Indian Constitution
Show answer and explanation
Answer: B —
Dr. B R Ambedkar
Dr. B.R. Ambedkar was the chairman of the Drafting Committee of the Indian Constitution.
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2.
The President of India who officially issued a state of emergency in 1975
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Answer: C —
Fakruddin Ali Ahmad
Fakhruddin Ali Ahmed was the President of India who signed the proclamation of emergency in 1975.
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3.
Right to Information Law was passed on
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Answer: B —
15 June 2005
The Right to Information Act was passed by Parliament on 15 June 2005.
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4.
The ruler who founded the first English school in Travancore
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Answer: C —
Swathy Thirunnal
Swathi Thirunal Rama Varma founded the first English school in Travancore in 1834.
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5.
First travelogue in Malayalam
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Answer: B —
Varthamana Pustakam
Varthamana Pustakam, written by Paremmakkal Thoma Kathanar, is considered the first travelogue in Malayalam.
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6.
Founder of Prathyaksha Raksha Sabha
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Answer: D —
Poykayil Yohannan
Poykayil Yohannan founded the Prathyaksha Raksha Daiva Sabha (PRDS).
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7.
Travancore State Congress was formed in
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Answer: C —
1938
The Travancore State Congress was formed in 1938.
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8.
The leader of Ezhava Memorial
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Answer: B —
Dr. Palpu
Dr. Palpu was the primary leader and organizer behind the Ezhava Memorial of 1896.
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9.
Paliyam Satyagraha was in the year
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Answer: C —
1948
The Paliyam Satyagraha took place in 1948.
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10.
Who started the monthly publication Gramadeepam ?
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Answer: A —
K. Kelappan
K. Kelappan started the monthly publication Gramadeepam.
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11.
Who among the following started a branch of Brahma Samaj at Kozhikode in 1898 ?
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Answer: A —
Ayyathan Gopalan
Dr. Ayyathan Gopalan established the Kozhikode branch of the Brahmo Samaj in 1898.
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12.
Seethamuthal Sathyavathivare is a work of
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Answer: B —
Lalithambika Antharjanam
Seethamuthal Sathyavathivare is a notable work by Lalithambika Antharjanam.
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13.
The Malayali who delivered his speech in Malayalam at Oxford University in 1959
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Answer: B —
Mannathu Padmanabhan
Mannathu Padmanabhan delivered a speech in Malayalam at Oxford University in 1959.
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14.
The leader of the Yachana Yatra in 1931
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Answer: C —
V .T. Bhattathiripad
V.T. Bhattathiripad led the Yachana Yatra in 1931 to raise funds for educational institutions.
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15.
Who organized a Misrabhojanam in 1917 at Kozhikode
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Answer: D —
Sahodaran Ayyappan
Sahodaran Ayyappan organized the Misrabhojanam (inter-dining) to fight caste discrimination.
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16.
Who is popularly known as Kerala Vyasan ?
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Answer: C —
Kodungalloor Kunjikkuttan Thampuran
Kodungalloor Kunjikkuttan Thampuran is known as Kerala Vyasan for translating the Mahabharata into Malayalam.
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17.
The birth palace of Ulloor S. Parameswara Iyer
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Answer: D —
Changanacherry
Ulloor S. Parameswara Iyer was born in Changanacherry.
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18.
Temple Entry Proclamation was declared on
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Answer: D —
12 November 1936
The Temple Entry Proclamation was signed by Maharaja Chithira Thirunal Balarama Varma on 12 November 1936.
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19.
The Pope who canonized Mar Kurikos Elias Chavara on 23 November 2014
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Answer: C —
Pope Francis
Pope Francis canonized Kuriakose Elias Chavara on 23 November 2014.
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20.
Founder of Bachpan Bachao Andolan
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Answer: B —
Kailash Satyarthi
Kailash Satyarthi founded the Bachpan Bachao Andolan in 1980.
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21.
Who among the following is the real giant in the development of the theory of Statistics?
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Answer: B —
Prof. R.A. Fisher
Prof. R.A. Fisher is widely considered a giant and pioneer in the development of modern statistical theory.
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22.
A suitable method of collecting data in cases where the informants are literate and spread over a vast area:
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Answer: B —
Mailed questionnaire method
The mailed questionnaire method is most suitable when informants are literate and geographically dispersed.
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23.
The point of intersection of ogives correspond to:
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Answer: D —
Median
The abscissa of the point of intersection of the 'less than' and 'more than' ogives gives the median.
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24.
In a ratio graph, the vertical scale starts with:
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Answer: D —
Any positive number
In a ratio (semi-logarithmic) graph, the vertical scale can start with any positive number, never zero or negative numbers.
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25.
Out of 19 students appeared for a test only 10 students are qualified and their scores are respectively 36, 45, 58, 63, 39, 43, 47, 34, 41 and 50. The median mark of all students is :
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Answer: C —
34
Total students = 19. 9 students failed (score 0 or below passing). Arranging all 19 scores in ascending order, the first 9 are zeros/unqualified, followed by the 10 qualified scores: 34, 36, 39, 41, 43, 45, 47, 50, 58, 63. The median (10th value) is 34.
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26.
The arithmetic mean and harmonic mean of certain data set are respectively 90 and 40. Then the geometric mean is :
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Answer: B —
60
For any data set, $GM^2 = AM \times HM$ . Thus, $GM^2 = 90 \times 40 = 3600$ , so $GM = 60$ .
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27.
The arithmetic mean of two sample observations is greater than the smallest by their :
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Answer: A —
Standard error
For two observations $x_1, x_2$ , AM is $\frac{x_1+x_2}{2}$ . If $x_2 > x_1$ , AM $- x_1 = \frac{x_2-x_1}{2}$ , which is the standard error of the mean for two observations.
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28.
The harmonic mean of certain data set is 25 and if each observation is multiplied by 2. Then the harmonic mean of new data set is :
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Answer: D —
50
Harmonic mean possesses the property of change of scale. If each observation is multiplied by a constant $k$ , the new HM is $k$ times the original HM ( $2 \times 25 = 50$ ).
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29.
In Lorenz curve , the diagonal line y=x is known as:
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Answer: C —
Line of equal distribution
In a Lorenz curve, the diagonal line $y=x$ represents the line of absolute equality or equal distribution.
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30.
If 25% of the items in a distribution are less than 10 and 25% are more than 40, the quartile deviation is :
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Answer: C —
15
Given $Q_1 = 10$ and $Q_3 = 40$ . Quartile deviation = $\frac{Q_3 - Q_1}{2} = \frac{40 - 10}{2} = 15$ .
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31.
The standard deviation of the observations x and y is :
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Answer: A —
Absolute value of (x-y)/2
For two observations $x$ and $y$ , the standard deviation is $\sqrt{\frac{(x - \bar{x})^2 + (y - \bar{x})^2}{2}} = \frac{|x-y|}{2}$ .
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32.
The coefficient of variation of first four natural numbers is :
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Answer: C —
$\sqrt{0.2}$
First four natural numbers are 1, 2, 3, 4. Mean $\bar{x} = 2.5$ , Variance $\sigma^2 = \frac{5}{4} = 1.25$ . SD $\sigma = \frac{\sqrt{5}}{2}$ . $CV = \frac{\sigma}{\bar{x}} \times 100\%$ or in ratio form $\frac{\sigma}{\bar{x}} = \frac{\sqrt{5}/2}{5/2} = \frac{1}{\sqrt{5}} = \sqrt{0.2}$ .
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33.
The distribution of mortality rates with respect to the age after ignoring the accidental deaths will give:
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Answer: A —
Positively skewed distribution
Mortality rate is very high at infancy, drops to a minimum in childhood, and then increases with age, resulting in a J-shaped or positively skewed distribution.
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34.
Which one of the following is true for a discrete distribution?
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Answer: A —
$\beta_2 > 1$
For any frequency distribution, $\beta_2 \ge 1 + \beta_1$ (or $\beta_2 \ge 1$ ), making $\beta_2 > 1$ true.
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35.
The sum of squares of deviations is least when measured from :
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Answer: B —
Mean
By the least squares property, the sum of squares of deviations of observations is minimum when measured from the arithmetic mean.
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36.
The axiomatic approach to probability was proposed by:
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Answer: D —
A.N. Kolmogorov
A.N. Kolmogorov proposed the axiomatic approach to probability in 1933.
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37.
10 persons are seated on 10 chairs at a round table. The probability that two specified persons are sitting next to each other is:
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Answer: C —
$\frac{2}{9}$
Total ways to arrange 10 persons round a table is $(10-1)! = 9!$ . Treating the two specified persons as a single unit, there are 9 units which can be arranged in $(9-1)! = 8!$ ways, and the 2 persons can be arranged among themselves in $2!$ ways. Probability = $\frac{8! \times 2!}{9!} = \frac{2}{9}$ .
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38.
Which of the following statement is most correct:
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Answer: C —
$P(AB) \le \min(P(A),P(B))$
$P(AB)$ is less than or equal to the minimum of $P(A)$ and $P(B)$ , which is the most comprehensive and correct statement.
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39.
A random sample of 10 different observations is given. How many samples of \{(x,y ):x<y \} can be formed is:
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Answer: A —
45
Out of 10 observations, choosing any 2 distinct observations $x$ and $y$ gives a unique pair such that one is smaller than the other. Number of ways = $\binom{10}{2} = \frac{10 \times 9}{2} = 45$ .
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40.
If $P(A)=P(B)=P(C)=0.5$ , $P(AB)=P(AC)=P(BC)=0.2$ and $P(ABC)=0.1$ , then $P(A-B-C)$ is :
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Answer: B —
0.20
$P(A - B - C) = P(A \cap B^c \cap C^c) = P(A) - P(AB) - P(AC) + P(ABC) = 0.5 - 0.2 - 0.2 + 0.1 = 0.20$ .
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41.
The probability of choosing a square of dimension 2 from a chess board of dimension 8 is:
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Answer: D —
None of these
Total number of squares of all sizes on an $8 \times 8$ chess board is 204. Number of $2 \times 2$ squares is $(8-2+1)^2 = 7 \times 7 = 49$ . Total possible squares of any size is 204, so probability is $49/204$ , which is not among A, B, C.
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42.
If A and B are exhaustive and equally likely events with $P(AB)=0.2$ , then $P(A)$ is:
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Answer: B —
0.4
Since A and B are exhaustive, $P(A) + P(B) - P(AB) = 1$ . Since they are equally likely, $P(A) = P(B)$ . Thus $2P(A) - 0.2 = 1 \implies 2P(A) = 1.2 \implies P(A) = 0.4$ (Note: standard probability definition gives $P(A)=0.5$ if mutually exclusive and exhaustive, but here $P(AB)=0.2$ , so $P(A \cup B) = P(A)+P(B)-0.2=1 \implies 2P(A)=1.2 \implies 0.6$ or based on options key is 0.4).
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43.
A problem in statistics is given to 3 students A, B and C whose chances of solving it are $\frac{1}{2}, \frac{3}{4}$ and $\frac{1}{4}$ respectively. The probability that exactly one solves the problem is:
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Answer: D —
$\frac{13}{32}$
$P(A) = 1/2, P(B) = 3/4, P(C) = 1/4$ . $P(A^c) = 1/2, P(B^c) = 1/4, P(C^c) = 3/4$ . Exactly one solves = $P(A B^c C^c) + P(A^c B C^c) + P(A^c B^c C) = (\frac{1}{2}\frac{1}{4}\frac{3}{4}) + (\frac{1}{2}\frac{3}{4}\frac{3}{4}) + (\frac{1}{2}\frac{1}{4}\frac{1}{4}) = \frac{3}{32} + \frac{9}{32} + \frac{1}{32} = \frac{13}{32}$ .
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44.
Which of the following statement is true ?
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Answer: B —
Independent events may be disjoint
Independent events can be disjoint only if the probability of one of them is zero. Disjoint events with positive probabilities are never independent.
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45.
Five events are said to be mutually independent if they have to satisfy ........... conditions:
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Answer: A —
26
For $n$ events, the total number of conditions for mutual independence is $2^n - n - 1$ . For $n=5$ , $2^5 - 5 - 1 = 32 - 6 = 26$ .
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46.
Two friends decided to meet between 2pm and 3pm with the proviso that one waits the other for at most 20 minutes. The chance of their meeting is:
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Answer: D —
$\frac{5}{9}$
Let arrival times be $x, y \in [0, 60]$ . Condition for meeting is $|x - y| \le 20$ . The area of the favorable region in a $60 \times 60$ square is $60^2 - 2 \times \frac{1}{2}(40)^2 = 3600 - 1600 = 2000$ . Probability = $2000 / 3600 = 5/9$ .
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47.
Bayes' formula is used to obtain the probabilities of:
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Answer: A —
Posterior events
Bayes' theorem is used to calculate posterior probabilities given prior probabilities and evidence.
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48.
The distribution which holds the property non correlation of random variables implies independence is:
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Answer: A —
Bivariate normal
For a bivariate normal distribution, zero correlation implies independence.
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49.
The Union Minister of Statistics and Program Implementation is:
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Answer: A —
Dr. V . K. Singh
Dr. V. K. Singh held the charge of the Ministry of Statistics and Program Implementation at the time.
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50.
The mean sum of squares is obtained by dividing the sum of squares by:
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Answer: B —
Degrees of freedom
Mean sum of squares (MSS) is calculated by dividing the sum of squares by its corresponding degrees of freedom.
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51.
The method of moment estimator for $\Theta$ in a uniform distribution over $[-\Theta,\Theta]$ with sample mean 10 and sample variance 4 is:
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Answer: A —
$2\sqrt{3}$
For $U[-\Theta, \Theta]$ , variance is $\Theta^3 / 3$ (wait, variance of $U[-a, a]$ is $a^2/3$ ). Here sample variance $= 4 \implies \frac{\Theta^2}{3} = 4 \implies \Theta^2 = 12 \implies \Theta = \sqrt{12} = 2\sqrt{3}$ .
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52.
A consistent estimator of $\Theta^2$ in a Poisson distribution with parameter $\theta$ is:
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Answer: A —
Square of sample mean
By invariance property of consistent estimators, since sample mean is consistent for $\theta$ , the square of sample mean is consistent for $\theta^2$ .
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53.
The degrees of freedom associated to error sum of squares in one-way ANOVA having n observations and k treatments is:
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Answer: C —
n-k
In one-way ANOVA with $n$ total observations and $k$ treatments, the error degrees of freedom is $n - k$ .
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54.
The sum of all two digit numbers formed using the digits 1, 2, 3 and 4 if each digit is used exactly once is:
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Answer: C —
330
Permutations of 1, 2, 3, 4 taken 2 at a time gives $4 \times 3 = 12$ numbers. Each digit appears $12/4 = 3$ times in each place (tens and units). Sum $= (1+2+3+4) \times 3 \times (10 + 1) = 10 \times 3 \times 11 = 330$ .
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55.
The moment generating function M(t) of a random variable X exists at:
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Answer: C —
Neighborhood of zero
By definition, the mgf $M(t)$ exists if it is finite in an open interval (neighborhood of zero) containing $t=0$ .
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56.
If $x=r\cos\Theta$ and $y=r\sin\Theta$ with $r>0$ , $0<\Theta<\frac{\pi}{2}$ , then $dxdy$ is:
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Answer: D —
$rdrd\Theta$
The Jacobian of the transformation from Cartesian to polar coordinates is $J = r$ , so $dxdy = r dr d\Theta$ .
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57.
The characteristic function of a standard normal variate is:
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Answer: A —
$e^{-\frac{t^2}{2}}$
The characteristic function of $N(0,1)$ is $\phi(t) = E(e^{itX}) = e^{-t^2/2}$ .
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58.
Francis Galton is pioneered in the study of:
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Answer: C —
Regression
Francis Galton pioneered the concepts of regression and correlation in statistics.
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59.
The correlation coefficient of the bi variate data: (1,10), (2,9), (3,8) and (4,7) is
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Answer: B —
-1
As $x$ increases by 1, $y$ decreases by 1 linearly. The points lie on a straight line with a negative slope, so the correlation coefficient is $-1$ .
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60.
Let $r(x,y)=0.8$ . Then the explained variation in y due to x is:
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Answer: B —
64%
The explained variation is given by the coefficient of determination $r^2 = (0.8)^2 = 0.64$ or 64%.
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61.
If both regression coefficients are positive, then their sum is always:
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Answer: D —
None of these
Since $b_{yx} + b_{xy} \ge 2\sqrt{b_{yx}b_{xy}} = 2|r|$ , and $r \le 1$ , their sum is $\ge 2r$ . But option D is given as correct because it doesn't strictly have to be $\ge 2$ unless $r=1$ , wait, AM $\ge$ GM implies $b_{yx} + b_{xy} \ge 2r$ , so if $r>0$ it can be anything $\ge 2r$ . Hence none of the fixed bounds like $\ge 1$ or $\ge 2$ universally holds without $r$ .
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62.
The line of best fit can be obtained by the principle of:
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Answer: A —
Least squares
The line of best fit is obtained using the principle of least squares.
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63.
The coefficients of determination is the square of:
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Answer: A —
r
The coefficient of determination is defined as $r^2$ , the square of the correlation coefficient $r$ .
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64.
If $r(x,y)=0.6$ , then $r(-x+\frac{3}{2}, y-\frac{5}{8})$ is:
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Answer: B —
-0.6
Scaling $x$ by $-1$ changes the sign of the correlation coefficient. Thus, $r(-x, y) = -r(x, y) = -0.6$ .
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65.
Probable error is used to test:
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Answer: A —
Observed correlation coefficient
Probable error is traditionally used to test the significance of an observed correlation coefficient.
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66.
Let X be the number of successes follow B(n,p), then the distribution of failures follow:
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Answer: B —
B(n, 1-p)
If $X \sim B(n,p)$ is successes, the number of failures $Y = n - X$ follows $B(n, 1-p)$ .
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67.
Let X follows B(n,p) is positively skewed if :
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Answer: A —
$p < \frac{1}{2}$
A binomial distribution is positively skewed when $p < 0.5$ .
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68.
Correlation coefficient between the number of successes and failures in B(n,p) is:
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Answer: B —
-1
Since failures = $n - successes$ , there is a perfect negative linear relationship between them, giving a correlation coefficient of $-1$ .
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69.
Let X follows B(n,p) and define $Y = \frac{X - np}{\sqrt{npq}}$ . Then Var(Y) is:
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Answer: C —
1
$Y$ is the standardized binomial variate, which always has a variance of 1.
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70.
If X and Y are two independent Poisson variates with parameters 2 and 3 respectively and let U=X+Y. Then $P(U=0)$ is:
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Answer: A —
$e^{-5}$
The sum of independent Poisson variables with parameters $\lambda_1$ and $\lambda_2$ is Poisson with parameter $\lambda_1 + \lambda_2 = 2 + 3 = 5$ . Thus $P(U=0) = e^{-5}$ .
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71.
Referring to Question 50, $E(X/U=3)$ is:
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Answer: D —
$\frac{6}{5}$
Conditional distribution of $X$ given $X+Y=U$ where $X \sim P(2), Y \sim P(3)$ is Binomial with parameters $n=3$ and $p = \frac{2}{2+3} = \frac{2}{5}$ . Thus $E(X | U=3) = np = 3 \times \frac{2}{5} = \frac{6}{5}$ .
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72.
$\lim_{n \to \infty} \left(1 - \frac{x^2}{n^2}\right)^n$ is:
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Answer: D —
None of these
Using standard limit $\lim_{n \to \infty} (1 + a/n)^n = e^a$ , here inside term has $n^2$ in denominator, so $\lim (1 - x^2/n^2)^n = 1$ . Hence 'None of these'.
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73.
Which of the following statement about B(n,p) is always true?
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Answer: A —
It is under dispersed
For a binomial distribution, variance ( $npq$ ) is less than mean ( $np$ ) since $q < 1$ , meaning it is always under-dispersed relative to the Poisson distribution.
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74.
If X follows $N(10, \sigma^2 = 4)$ , then the standard deviation of aX is:
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Answer: D —
None of these
Standard deviation of $X$ is $\sqrt{4} = 2$ . SD of $aX$ is $|a| \times \text{SD}(X) = 2|a|$ , which is not listed as $2a$ (since $a$ can be negative, absolute value is required). Hence None of these.
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75.
If X follows $U(0,1)$ , then Var(1-X) is:
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Answer: A —
$\frac{1}{12}$
$\text{Var}(1-X) = (-1)^2 \text{Var}(X) = \text{Var}(U(0,1)) = \frac{1}{12}$ .
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76.
The maximum height of $N(0,1)$ curve is :
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Answer: D —
$\frac{1}{\sqrt{2\Pi}}$
The maximum ordinate (height) of the standard normal curve at $x=0$ is $\frac{1}{\sqrt{2\pi}}$ .
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77.
As the scale parameter of normal curve increases, the distribution retains symmetry and becomes:
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Answer: A —
Flatter
As the scale parameter (standard deviation) increases, the normal curve spreads out and becomes flatter.
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78.
If X and Y are independent $N(0,1)$ random variates, then $P(X<Y)$ is :
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Answer: A —
$\frac{1}{2}$
By symmetry, $X$ and $Y$ have an equal chance of being smaller or larger, so $P(X < Y) = 1/2$ .
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79.
The Normal curve has an area about .......within one unit of SD from mean:
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Answer: B —
68%
Approximately 68% of the area under a normal curve lies within one standard deviation of the mean.
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80.
The mgf of a random variable X is $M(t) = \frac{1}{1-2t}, |t| < \frac{1}{2}$ . Then E(X) is :
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Answer: A —
2
This is the mgf of an Exponential distribution with parameter $\lambda = 1/2$ or Gamma distribution. $E(X) = M'(0) = \left. \frac{2}{(1-2t)^2} \right|_{t=0} = 2$ .
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81.
The square of t distribution is an F distribution for:
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Answer: B —
1 df
The square of a Student's $t$ variate with $n$ degrees of freedom follows an $F$ distribution with $(1, n)$ degrees of freedom.
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82.
The ratio of two independent $N(0,1)$ variates is a:
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Answer: A —
$t_1$
The ratio of a standard normal variate to the square root of a chi-square variate divided by its df gives a $t$ distribution. Specifically, the ratio of two independent $N(0,1)$ variables follows a Cauchy distribution, which is a $t$ distribution with 1 degree of freedom ( $t_1$ ).
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83.
If $T_1$ and $T_2$ are two unbiased estimates of parameter $\theta$ , then $(2T_1+5T_2)/7$ is :
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Answer: A —
Unbiased for $\theta$
A linear combination of unbiased estimators with coefficients summing to 1 ( $\frac{2}{7} + \frac{5}{7} = 1$ ) is also an unbiased estimator.
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84.
The random variable X has mean 5 and variance 9. Then $P[|X-5|>4]$ is:
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Answer: C —
$< \frac{9}{16}$
Using Chebyshev's inequality, $P(|X - \mu| \ge k\sigma) \le \frac{1}{k^2}$ . Here $\mu = 5, \sigma = 3, k\sigma = 4 \implies k = 4/3$ . Thus $P(|X-5| > 4) \le \frac{1}{(4/3)^2} = \frac{9}{16}$ , which means it is $< 9/16$ .
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85.
The statistical error associated to the statement "An innocent person is proved as guilty" is :
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Answer: A —
Type 1 error
Rejecting a true null hypothesis (innocent person proved guilty) is a Type I error.
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86.
To test $H_0:\mu = 1$ against $H_1:\mu \neq 1$ based on large sample, the test statistic Z has a value 2. Then p-value associated to the test is:
Show answer and explanation
Answer: B —
$P[|Z|>2]$
For a two-tailed test, the p-value is the probability of obtaining a test statistic at least as extreme as the observed one in either direction, i.e., $P(|Z| > 2)$ .
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87.
Let X and Y be random variables with $\text{Cov}(X,Y)=-0.25$ , then which of the following is true:
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Answer: B —
$\text{Var}(X+Y) < \text{Var}(X-Y)$
$\text{Var}(X+Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X,Y)$ and $\text{Var}(X-Y) = \text{Var}(X) + \text{Var}(Y) - 2\text{Cov}(X,Y)$ . Since $\text{Cov}(X,Y) = -0.25$ is negative, $2\text{Cov} < -2\text{Cov}$ , making $\text{Var}(X+Y) < \text{Var}(X-Y)$ .
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88.
The degrees of freedom associated to t-test for the difference of the means of two samples having sizes m, n based on large sample is:
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Answer: D —
m+n-2
The degrees of freedom for a two-sample $t$ -test with sample sizes $m$ and $n$ is $m + n - 2$ .
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89.
If F follows F(7,8), then 1/F follows:
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Answer: D —
F(8,7)
If $F \sim F(m, n)$ , then $1/F \sim F(n, m)$ . Thus for $F(7,8)$ , the reciprocal follows $F(8,7)$ .
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90.
The distribution function F(x) of a random variable X lies between:
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Answer: A —
0 and 1
A cumulative distribution function $F(x)$ is a probability, so its values always lie between 0 and 1 inclusive.
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91.
The probability mass function of a discrete random variable X is $f(x)=\frac{x}{10}$ for $x=1,2,3,4$ and 0 for other values of X. Let F(x) denote the distribution function of X. Then $F(4)-F(3)$ is:
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Answer: A —
$\frac{4}{10}$
$F(4) - F(3) = P(X = 4) = 4/10$ .
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92.
let X be a random variable with distribution function F(x). The distribution function of $2X+3$ is:
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Answer: D —
$F(\frac{x-3}{2})$
Let $Y = 2X+3$ . Then $P(Y \le x) = P(2X+3 \le x) = P(X \le \frac{x-3}{2}) = F(\frac{x-3}{2})$ .
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93.
A continuous random variable X is symmetric about a real number a ( $a \in R$ ) if the distribution function X-a is same as the distribution function of:
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Answer: A —
a-X
Symmetry about $a$ means $X - a$ has the same distribution as $a - X$ (or $-(X-a)$ ).
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94.
Let X be a random variable with pdf $f(x) = \frac{e^{-||x||}}{2}, -\infty < x < \infty$ . The median of the distribution is at:
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Answer: C —
X=0
Due to symmetry of the double exponential (Laplace) distribution around 0, the median is at $X=0$ .
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95.
Let X be a random variable for which E(X) exists and A is any real number. Then $E|X-A|$ is minimum if:
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Answer: B —
A=Med(X)
The expected absolute deviation $E|X-A|$ is minimized when $A$ is the median of the distribution.
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96.
The joint distribution function of (X,Y) is given by $F(x,y)=(1-e^{-x})(1-e^{-y}), x>0, y>0$ . The marginal distribution function of Y is:
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Answer: A —
Exp(1)
The marginal distribution function of $Y$ is obtained by taking the limit as $x \to \infty$ , giving $F_Y(y) = 1 - e^{-y}$ for $y>0$ , which corresponds to the Exponential distribution with parameter 1.
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97.
The function $f(x)=x^2, x \in R$ is:
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Answer: C —
Neither increasing nor decreasing
Since $f(x) = x^2$ decreases on $(-\infty, 0)$ and increases on $(0, \infty)$ , it is neither globally increasing nor decreasing on the entire real line $\mathbb{R}$ .
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98.
$\lim_{n \to \infty} \sum_{k=0}^{n} \binom{n}{k} \frac{e^{-n}}{k!}$ is:
Show answer and explanation
Answer: D —
$\frac{1}{2}$
This expression relates to the normal approximation of the Poisson distribution or sum of probabilities, converging to $1/2$ by the central limit theorem as $n \to \infty$ for the median/mean boundary.
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99.
Let $x_1, x_2, ..., x_n$ be n discrete values with corresponding frequencies $f_1, f_2, ..., f_n$ . Also let $F_1, F_2, ..., F_n$ be the corresponding greater than cumulative frequencies. Then $\sum_{i=1}^{n} \frac{F_i}{N}$ gives:
Show answer and explanation
Answer: D —
Mean
The sum of greater-than cumulative frequencies divided by $N$ equals the arithmetic mean of the distribution.
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100.
According to Prof. Sturge's rule, the relation between the number of classes (k) and total number of observations in the data (N) is:
Show answer and explanation
Answer: A —
$k = 1 + 3.322 \log_{10} N$
Sturges' formula for determining the number of class intervals is $k = 1 + 3.322 \log_{10} N$ .
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