Discrete random variables; expected value practice questions.

A discrete random variable X has P(X = x) = kx for x = 1, 2, 3, 4. Determine the value of k.

• 1/4
• 1/3
• 1/10
• 10

Answer: 1/10

The probabilities must sum to 1: k(1 + 2 + 3 + 4) = 10k = 1, so k = 1/10.

A game costs $3 to play. With probability 0.2 the player wins $10; otherwise the player wins nothing. Which statement best describes the game in the long run?

• The player loses $1 per game on average
• The player gains $1 per game on average
• The game is fair
• The player loses $2 per game on average

Answer: The player loses $1 per game on average

Expected gain = 0.2 × 10 − 3 = 2 − 3 = −1, so on average the player loses $1 per game.

X takes the values 1, 2 and 3 with P(X = 1) = 0.2 and P(X = 3) = 0.5. Calculate E(X).

• 2.0
• 1.8
• 2.5
• 2.3

Answer: 2.3

First P(X = 2) = 1 − 0.2 − 0.5 = 0.3, then E(X) = 1(0.2) + 2(0.3) + 3(0.5) = 2.3.

A random variable takes the values 1, 2, 3, 4. Which set of probabilities, listed in that order, forms a valid probability distribution?

• 0.2, 0.3, 0.4, 0.2
• 0.1, 0.2, 0.3, 0.4
• 0.5, 0.5, 0.1, −0.1
• 0.3, 0.3, 0.2, 0.1

Answer: 0.1, 0.2, 0.3, 0.4

Each probability must lie between 0 and 1 and they must sum to exactly 1; only 0.1 + 0.2 + 0.3 + 0.4 = 1 satisfies both conditions.

Two fair coins are tossed and X is the number of heads obtained. Determine E(X).

• 0.5
• 2
• 1
• 1.5

Answer: 1

P(X = 0) = 1/4, P(X = 1) = 1/2, P(X = 2) = 1/4, so E(X) = 0(1/4) + 1(1/2) + 2(1/4) = 1.

A raffle sells 200 tickets at $2 each. There is a single prize worth $150. Calculate the expected net gain, in dollars, for a person who buys one ticket.

• −1.25
• −0.75
• 0.75
• −2

Answer: −1.25

Expected winnings = 150/200 = $0.75; subtracting the $2 ticket price gives an expected gain of 0.75 − 2 = −$1.25.