Probability MCQs

1

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).

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

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).

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

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).

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

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.

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

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).

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

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?

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

Mathematics • Probability

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

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

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?

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

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).

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

Mathematics • Probability

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

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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