Probability MCQs

1

Advanced Mathematics • Probability

Consider 10 tosses of a fair die. Let X_i denote the number of times digit i appears. Find P(X_2 = 2 | X_1 = 2). Consider 10 tosses of a fair die. Let X_i denote the number of times digit i appears. Find P(X_2 = 2 | X_1 = 2).

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A. C(10,2)(1/6)^2(5/6)^8 C(10,2)(1/6)^2(5/6)^8 B. C(8,2)(1/5)^2(4/5)^6 C(8,2)(1/5)^2(4/5)^6 C. C(8,2)(1/6)^2(5/6)^6 C(8,2)(1/6)^2(5/6)^6 D. (1/6)^2 (1/6)^2

सही उत्तर: B Correct Answer: B

Given X_1 = 2, the remaining 8 outcomes are non-one. For each such outcome, the conditional probability of getting 2 is 1/5. Hence X_2 is binomial with n = 8 and p = 1/5.

2

Advanced Mathematics • Probability

We have P(X = x, Y = y) = 1/9 for all x, y in {1, 2, 3}. Find P(X > Y). We have P(X = x, Y = y) = 1/9 for all x, y in {1, 2, 3}. Find P(X > Y).

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A. 0 0 B. 1/3 1/3 C. 1 1 D. 1/2 1/2

सही उत्तर: B Correct Answer: B

The ordered pairs satisfying X > Y are (2,1), (3,1), and (3,2). Their total probability is 3 x (1/9) = 1/3.

3

Advanced Mathematics • Probability

Consider 10 tosses of a biased coin with probability of obtaining a head equal to 1/3. Let Y denote the total number of tails observed. Find E(5Y + 2). Consider 10 tosses of a biased coin with probability of obtaining a head equal to 1/3. Let Y denote the total number of tails observed. Find E(5Y + 2).

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A. 10 10 B. 56/3 56/3 C. 106/3 106/3 D. 10/3 10/3

सही उत्तर: C Correct Answer: C

The probability of a tail is 2/3, so E(Y) = 10 x 2/3 = 20/3. Thus E(5Y + 2) = 5E(Y) + 2 = 106/3.

4

Advanced Mathematics • Probability

The density of a random variable X is f_X(x) = a + bx^2 for x in [0,1], and 0 otherwise. Find (a,b) if E(X) = 3/5. The density of a random variable X is f_X(x) = a + bx^2 for x in [0,1], and 0 otherwise. Find (a,b) if E(X) = 3/5.

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A. (3/5, -3/5) (3/5, -3/5) B. (3, -3) (3, -3) C. (1, 2) (1, 2) D. (3/5, 6/5) (3/5, 6/5)

सही उत्तर: D Correct Answer: D

Normalization gives a + b/3 = 1, while E(X) = a/2 + b/4 = 3/5. Solving yields a = 3/5 and b = 6/5.

5

Advanced Mathematics • Probability

Suppose (X,Y) is jointly distributed with density f(s,t) = 1/pi if s^2 + t^2 <= 1, and 0 otherwise. Find Cov(X,Y). Suppose (X,Y) is jointly distributed with density f(s,t) = 1/pi if s^2 + t^2 <= 1, and 0 otherwise. Find Cov(X,Y).

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A. -1 -1 B. 1 1 C. 0 0 D. pi pi

सही उत्तर: C Correct Answer: C

By symmetry of the uniform distribution on the unit disk, E(X)=E(Y)=0 and E(XY)=0. Hence Cov(X,Y)=0.

6

Advanced Mathematics • Probability

Which one of the following gives a pair of independent random variables? Which one of the following gives a pair of independent random variables?

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A. 10 coins are thrown at random. X denotes the total number of heads and Y denotes the total number of tails. 10 coins are thrown at random. X denotes the total number of heads and Y denotes the total number of tails. B. 10 coins are tossed at random. X denotes the total number of heads on even tosses (2nd, 4th, ..., 10th) and Y denotes the total number of tails on odd tosses (1st, 3rd, ..., 9th). 10 coins are tossed at random. X denotes the total number of heads on even tosses (2nd, 4th, ..., 10th) and Y denotes the total number of tails on odd tosses (1st, 3rd, ..., 9th). C. For 1 <= i <= 6, let X_i denote the number of times digit i appears in 10 tosses of a die. Consider (X_1,X_2). For 1 <= i <= 6, let X_i denote the number of times digit i appears in 10 tosses of a die. Consider (X_1,X_2). D. Two dice are thrown at random; X is the maximum and Y is the minimum of the digits appearing. Two dice are thrown at random; X is the maximum and Y is the minimum of the digits appearing.

सही उत्तर: B Correct Answer: B

X depends only on the even-numbered tosses and Y only on the disjoint odd-numbered tosses. Since all tosses are independent, X and Y are independent.

7

Advanced Mathematics • Probability

Which one of the following statements is true? Which one of the following statements is true?

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A. For two random variables X and Y, Cov(X,Y)=0 implies that X and Y are independent. For two random variables X and Y, Cov(X,Y)=0 implies that X and Y are independent. B. If V(X)=0, then P(X=0)=1. If V(X)=0, then P(X=0)=1. C. Let F be the distribution function of a random variable. Then lim as n approaches infinity of F(x + 1/n) always exists. Let F be the distribution function of a random variable. Then lim as n approaches infinity of F(x + 1/n) always exists. D. If P(X <= 100)=1, then E(X) always exists. If P(X <= 100)=1, then E(X) always exists.

सही उत्तर: C Correct Answer: C

A distribution function is right-continuous, so F(x+1/n) approaches F(x). The other statements are not true in general.

8

Advanced Mathematics • Probability

What is the probability that, in a random arrangement of the English alphabet, the word MOTHER will appear? What is the probability that, in a random arrangement of the English alphabet, the word MOTHER will appear?

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A. (1/6)^3 (1/6)^3 B. 20!/26! 20!/26! C. 21!/26! 21!/26! D. 22!/26! 22!/26!

सही उत्तर: C Correct Answer: C

Treat MOTHER as one fixed block. Together with the other 20 letters there are 21 objects, giving 21! favourable arrangements out of 26! total arrangements.

9

Advanced Mathematics • Probability

Let X and Y be two independent N(0,1) random variables. Find Cov(X+3Y, 5X-2Y). Let X and Y be two independent N(0,1) random variables. Find Cov(X+3Y, 5X-2Y).

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A. 1 1 B. -1 -1 C. 0 0 D. 11 11

सही उत्तर: B Correct Answer: B

Using independence and unit variances, Cov(X+3Y,5X-2Y)=5Var(X)-6Var(Y)=5-6=-1.

10

Advanced Mathematics • Probability

Which one of the following statements is false? Which one of the following statements is false?

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A. Two independent events cannot be disjoint. Two independent events cannot be disjoint. B. For a probability density function f, we always have |f(x)|<=1 for all real x. For a probability density function f, we always have |f(x)|<=1 for all real x. C. For an N(0,2) random variable, mean=median=mode. For an N(0,2) random variable, mean=median=mode. D. Any continuous non-negative function with integral from minus infinity to infinity equal to 1 gives a density function. Any continuous non-negative function with integral from minus infinity to infinity equal to 1 gives a density function.

सही उत्तर: B Correct Answer: B

A probability density can exceed 1 on a sufficiently short interval; only its integral over the whole line must equal 1. Therefore option B is false.

Probability MCQs

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