Random Variables, Expectation and Variance

Which of the following is a continuous random variable?

1. aThe number of heads in 5 tosses of a coin
2. bThe number of customers entering a shop in an hour
3. cThe height of a student chosen at random
4. dThe number of defective bulbs in a box of 10

A random variable \(X\) takes the values 0, 1, 2 and 3 with probabilities 0.1, \(k\), 0.3 and \(2k\) respectively. What is the value of \(k\)?

1. a0.2
2. b0.3
3. c0.6
4. d0.15

A random variable \(X\) takes the values 1, 2 and 3, with \(P(X = x) = kx\). What is \(P(X \ge 2)\)?

1. a\(\dfrac{1}{2}\)
2. b\(\dfrac{1}{3}\)
3. c\(\dfrac{1}{6}\)
4. d\(\dfrac{5}{6}\)

Three fair coins are tossed, and \(X\) is the number of heads. What is \(P(X = 2)\)?

1. a\(\dfrac{1}{8}\)
2. b\(\dfrac{3}{8}\)
3. c\(\dfrac{1}{2}\)
4. d\(\dfrac{2}{3}\)

A random variable \(X\) takes the values 0, 1, 2 and 3 with probabilities 0.2, 0.4, 0.3 and 0.1 respectively. What is \(E(X)\)?

1. a1.5
2. b1
3. c1.3
4. d0.25

A random variable \(X\) takes the values 1, 2 and 3 with probabilities 0.25, 0.5 and 0.25 respectively. What is the variance of \(X\)?

1. a4.5
2. b0.5
3. c2.5
4. d2

A random variable \(X\) takes the values 0, 2 and 4 with probabilities \(\dfrac{1}{4}\), \(\dfrac{1}{2}\) and \(\dfrac{1}{4}\) respectively. What is the standard deviation of \(X\)?

1. a2
2. b6
3. c\(\sqrt{6}\)
4. d\(\sqrt{2}\)

A fair die is thrown twice, and \(X\) is the number of sixes obtained. What is \(E(X)\)?

1. a\(\dfrac{1}{3}\)
2. b\(\dfrac{1}{6}\)
3. c\(\dfrac{11}{36}\)
4. d\(\dfrac{1}{2}\)

A bag contains 2 red and 3 black balls. Two balls are drawn at random without replacement, and \(X\) is the number of red balls drawn. What is \(E(X)\)?

1. a1
2. b0.4
3. c0.8
4. d1.2

For a random variable \(X\), \(E(X) = 3\) and \(E(X^{2}) = 13\). What is the standard deviation of \(X\)?

1. a4
2. b2
3. c\(\sqrt{13}\)
4. d\(\sqrt{10}\)

Assertion (A): The number of heads obtained when a coin is tossed 4 times is a discrete random variable.

Reason (R): A discrete random variable can take only a countable number of separate values.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): The table in which \(X\) takes the values 0, 1 and 2 with probabilities 0.4, 0.5 and 0.3 is a probability distribution.

Reason (R): In a probability distribution, each probability lies between 0 and 1, and all the probabilities add up to 1.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Assertion (A): If \(X\) is the number shown when a fair die is thrown, then \(E(X) = 3.5\).

Reason (R): The expected value of a random variable is always one of the values that the variable can take.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

1. I. The time taken by a runner to finish a race is a continuous random variable.
2. II. The number of students absent from a class on a given day is a continuous random variable.
3. III. A random variable assigns a real number to each outcome of a random experiment.

Which of the statements above is/are correct?

1. aOnly I and III
2. bOnly I
3. cOnly II and III
4. dI, II and III

1. I. \(E(X) = 1.7\)
2. II. \(P(X \le 2) = 0.8\)
3. III. \(\text{Var}(X) = 3.5\)

A random variable \(X\) takes the values 1, 2 and 3 with probabilities 0.5, 0.3 and 0.2 respectively. Which of the given statements about \(X\) is/are correct?

1. aOnly I
2. bOnly II
3. cOnly II and III
4. dOnly I and II

1. I. \(\text{Var}(X) = E(X^{2}) - [E(X)]^{2}\)
2. II. The standard deviation of \(X\) is the positive square root of its variance.
3. III. If \(X\) takes the value 5 with probability 1, then \(\text{Var}(X) = 0\).

Which of the statements above about a random variable \(X\) is/are correct?

1. aOnly I and II
2. bI, II and III
3. cOnly I
4. dOnly II and III

Two fair coins are tossed, and \(X\) is the number of heads, so \(X\) takes the values 0, 1 and 2 with probabilities \(\dfrac{1}{4}\), \(\dfrac{1}{2}\) and \(\dfrac{1}{4}\). Match List I with List II:

1. a(i)-R, (ii)-S, (iii)-P, (iv)-Q
2. b(i)-R, (ii)-P, (iii)-S, (iv)-Q
3. c(i)-S, (ii)-R, (iii)-P, (iv)-Q
4. d(i)-R, (ii)-S, (iii)-Q, (iv)-P

Match List I with List II (here \(\mu = E(X)\)):

1. a(i)-S, (ii)-Q, (iii)-R, (iv)-P
2. b(i)-Q, (ii)-S, (iii)-R, (iv)-P
3. c(i)-Q, (ii)-S, (iii)-P, (iv)-R
4. d(i)-S, (ii)-Q, (iii)-P, (iv)-R

At a school fair, Riya pays ₹ 10 to throw a fair die once. She wins ₹ 30 if the die shows 6, ₹ 9 if it shows 4 or 5, and nothing otherwise. What is her expected gain or loss per game?

1. aA gain of ₹ 8
2. bA loss of ₹ 3.50
3. cA gain of ₹ 2
4. dA loss of ₹ 2

Over 50 days, a shopkeeper recorded the number of laptops sold each day: 0 laptops on 5 days, 1 on 15 days, 2 on 15 days, 3 on 10 days and 4 on 5 days. He makes a profit of ₹ 2,000 on each laptop. Using these records, what is his expected daily profit?

1. a₹ 4,000
2. b₹ 38,000
3. c₹ 3,800
4. d₹ 2,000