Topic 4.7: Discrete Probability Distributions Questions

State what E(X) represents in practical terms.

A discrete random variable has P(X=2) = 0.3, P(X=4) = 0.5, P(X=6) = 0.2. Find P(X ≤ 4).

A fair six-sided die is rolled. Find E(X), the expected score.

A discrete random variable X has P(X=1) = 0.2, P(X=2) = 0.5, P(X=3) = k. Find k.

Find E(X) for the distribution P(X=0) = 0.4, P(X=1) = 0.4, P(X=2) = 0.2.

A spinner scores X with P(X=1) = 0.5, P(X=3) = 0.3, P(X=10) = 0.2. Find the expected score and explain why the spinner rarely shows the expected value.

For a distribution with E(X) = 2.4, a game pays out X dollars and costs $2.40 to play. State whether the game is fair.

A random variable X has Var(X) = 2.5.

(a) Find Var(4X − 3).

(b) The random variable Y is independent of X and has Var(Y) = 1.5. Find Var(X + Y) and the standard deviation of X + Y.

A discrete random variable X has the probability distribution shown: P(X=1) = 0.1, P(X=2) = 0.3, P(X=3) = k, P(X=4) = 0.2.

In a fairground game, a player spins a wheel and wins a prize X dollars. The distribution is P(X=0) = 0.5, P(X=1) = 0.3, P(X=5) = 0.2. It costs $2 to play.

The number of goals X scored by a football team in a match has the probability distribution below, where k is a constant.

x: 1, 2, 3, 4

P(X = x): 0.1, 0.4, k, 0.2

(a) Find the value of k.

(b) Write down the most likely number of goals (the mode).

(c) Find E(X), and hence find P(X > E(X)).

A discrete random variable X takes values 1, 2 and 3 with P(X=1) = 0.5, P(X=2) = 0.3, P(X=3) = 0.2. The experiment is performed twice independently.

At a small bakery, the number of loaves X a customer buys has the distribution below.

x: 0, 1, 2, 3

P(X = x): 0.2, 0.5, 0.2, 0.1

(a) Find E(X).

(b) Find the variance and standard deviation of X, giving each answer to three significant figures.

A raffle ticket has a net winning X dollars (a loss is negative). The distribution is:

x: −5, 10, w

P(X = x): 0.7, 0.25, 0.05

where w is the value of the top prize. The game is fair, so E(X) = 0. Find the value of the top prize w.