All subjects · Mathematics · WST01 (S1)

Variance, unknown probabilities and functions of X

Building on expectation, this lesson covers the harder exam skills: finding unknown probabilities from E(X), calculating variance, finding E(g(X)), and combining two independent observations.

Unknowns from two equations

If a table has two unknowns, use:

1. Σ P(X = x) = 1

2. the given E(X) = Σ x P(X = x)

Then solve simultaneously, showing the working (examiners require the steps, not just calculator output).

For the table: a + b = 0.7 and 0.1 + 2a + 3b + 0.8 = 2.6, so 2a + 3b = 1.7, giving b = 0.3 and a = 0.4.

x1234
P(X = x)0.1ab0.2

Variance and E(g(X))

E(g(X)) = Σ g(x) P(X = x) for any function g. So E(X2) = Σ x2 P(X = x) and E(1/X) = Σ (1/x) P(X = x).

Var(X) = E(X2) - [E(X)]2. Then Var(aX + b) = a2 Var(X): the sign of a disappears, so Var(5 - 2X) = 4 Var(X), never 5 - 2Var(X).

Two independent observations

If X1 and X2 are independent observations of X, find P(X1 + X2 = 3) by listing the ways: (1, 2) or (2, 1), so 2 × P(X = 1) × P(X = 2). Use probabilities from p(x), not values of F(x).

Worked example

With a = 0.4 and b = 0.3 in the table, find Var(X) and Var(5 - 2X).

  1. E(X) = 0.1 + 0.8 + 0.9 + 0.8 = 2.6
  2. E(X2) = 0.1 + 1.6 + 2.7 + 3.2 = 7.6
  3. Var(X) = 7.6 - 2.62 = 0.84

Answer: Var(X) = 0.84; Var(5 - 2X) = 4 × 0.84 = 3.36

Key idea: Use ΣP = 1 and E(X) to find two unknowns; E(g(X)) = Σg(x)P(X = x); Var(aX + b) = a2Var(X).

Check yourself

  1. In the table, E(X) = 2.6. Find a.
    Show answer

    0.4. a + b = 0.7 and 2a + 3b = 1.7. Then 2a + 3(0.7 - a) = 1.7, so -a = -0.4 and a = 0.4.

  2. X takes 1, 2, 3, 4 with probabilities 0.1, 0.4, 0.3, 0.2. Find E(X2).
    Show answer

    7.6. 1(0.1) + 4(0.4) + 9(0.3) + 16(0.2) = 7.6.

  3. X takes 1, 2, 3, 4 with probabilities 0.1, 0.4, 0.3, 0.2. Find Var(X).
    Show answer

    0.840. E(X) = 2.6, E(X2) = 7.6, so Var(X) = 7.6 - 6.76 = 0.84.

  4. Var(X) = 0.84. Find Var(5 - 2X).
    Show answer

    3.36. (-2)2 × 0.84 = 3.36.

  5. X takes 1, 2, 3, 4 with probabilities 0.1, 0.4, 0.3, 0.2. Find E(1/X).
    Show answer

    0.45. 0.1/1 + 0.4/2 + 0.3/3 + 0.2/4 = 0.45.

  6. X takes 1, 2, 3, 4 with probabilities 0.1, 0.4, 0.3, 0.2. Two independent observations X1 and X2 are taken. Find P(X1 + X2 = 3).
    Show answer

    0.08. (1, 2) or (2, 1): 2 × 0.1 × 0.4 = 0.08.

  7. E(X) = 2.6. Find E(5 - 2X).
    Show answer

    -0.2. 5 - 2(2.6) = -0.2.

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Last updated 7 November 2026. Spotted a mistake? Tell us.