H2 Maths 6.2 - Custom PMF: Finding E(X) and Var(X)

Step 1 - Read the probability table

A discrete random variable X has the following probability mass function.

Step 1 - Read the probability table

The values of X are zero, one, two, and three, with probabilities zero point one, zero point three, zero point four, and k.

Step 1 - Read the probability table

Our first job is to find the unknown probability k.

Step 2 - Find the unknown probability

All probabilities must sum to one.

Step 2 - Find the unknown probability

So zero point one plus zero point three plus zero point four plus k equals one.

Step 2 - Find the unknown probability

That gives zero point eight plus k equals one.

Step 2 - Find the unknown probability

Therefore k equals zero point two.

Step 3 - Compute E(X)

E of X equals the sum of each value times its probability.

Step 3 - Compute E(X)

That is zero times zero point one, plus one times zero point three, plus two times zero point four, plus three times zero point two.

Step 3 - Compute E(X)

Working through: zero plus zero point three plus zero point eight plus zero point six.

Step 3 - Compute E(X)

E of X equals one point seven.

Step 4 - Compute E(X²)

For the variance we need E of X squared.

Step 4 - Compute E(X²)

Square each value before multiplying by its probability.

Step 4 - Compute E(X²)

Zero squared times zero point one, plus one squared times zero point three, plus four times zero point four, plus nine times zero point two.

Step 4 - Compute E(X²)

That gives zero plus zero point three plus one point six plus one point eight.

Step 4 - Compute E(X²)

E of X squared equals three point seven.

Step 5 - Compute Var(X) and state the answer

Variance equals E of X squared minus E of X all squared.

Step 5 - Compute Var(X) and state the answer

That is three point seven minus one point seven squared.

Step 5 - Compute Var(X) and state the answer

One point seven squared is two point eight nine.

Step 5 - Compute Var(X) and state the answer

So the variance of X is zero point eight one.

Step 5 - Compute Var(X) and state the answer

Common mistake: forgetting to square the mean. Always subtract the square of E of X, not E of X itself.