Discrete Random Variables

Worked example

Coin flipped 3 times.

Let X=number of heads.

What are possible values?

Solution

1. All outcomes: HHH, HHT, HTH, HTT, THH, THT, TTH, TTT
2. X can be: 0 heads (TTT), 1 head (3 ways), 2 heads (3 ways), 3 heads (HHH)
3. X ∈ {0,1,2,3} - discrete random variable

Worked example

Coin 3 times, X=heads.

Find P(X=0), P(X=1), P(X=2), P(X=3).

Solution

1. P(X=0)=1/8 (1 way)
2. P(X=1)=3/8 (3 ways)
3. P(X=2)=3/8 (3 ways)
4. P(X=3)=1/8 (1 way)
5. Total: 1/8+3/8+3/8+1/8=1 ✓

Worked example

From coin example: find E(X) and Var(X).

Solution

1. E(X)=0(1/8)+1(3/8)+2(3/8)+3(1/8)
2. E(X)=0+3/8+6/8+3/8=12/8=1.5
3. E(X²)=0²(1/8)+1²(3/8)+2²(3/8)+3²(1/8)=15/8
4. Var(X)=15/8-1.5²=15/8-2.25=0.75

IB-style question — find a value from E(X) [5 marks]

In a board game, a player moves a token and the score X (in points) has the distribution below, where n is a positive whole number.

x: −2, 1, n

P(X = x): 0.4, 0.45, 0.15

The expected score is E(X) = 1.

(a) Show that the probabilities given are consistent with a valid distribution.

(b) Find the value of n.

Step by step

1. (a) Check the probabilities add to 1.

2. They sum to 1 and each lies between 0 and 1, so the distribution is valid.

3. (b) Write the expected value with n still unknown and set it equal to 1.

4. Simplify the known terms.

5. Collect and solve for n.

IB-style question — is the game fair?

A game costs $2 to play. You win $5 (net gain $3) with probability 0.3, otherwise you lose your $2.

(a) Find the expected gain per play. (b) Is the game fair? (c) Find the expected number of wins in 50 plays.

Step by step

1. (a) E(gain) = Σ (gain × probability).

2. (b) A fair game has E = 0. Here E = −$0.50, so it is NOT fair (it favours the house).

3. (c) Expected number of wins = n × P(win) (this is np, not E(gain)).

Worked example

E(X)=1.5, Var(X)=0.75.

Find E(2X+5) and Var(2X+5).

Solution

1. E(2X+5)=2E(X)+5=2(1.5)+5=8
2. Var(2X+5)=2²Var(X)=4(0.75)=3