Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — expectation and variance from a table
X takes values 1,2,3,4 with probabilities 0.1,0.2,0.4,0.3. Find E(X) and Var(X).
- 1Check the probabilities sum to 1: 0.1+0.2+0.4+0.3=1 ✓Always verify before calculating.
- 2E(X)=∑xP(X=x)=0.1+0.4+1.2+1.2Each value times its probability.
- 3E(X)=2.9Between 1 and 4, as it must be.
- 4E(X2)=1(0.1)+4(0.2)+9(0.4)+16(0.3)=9.3Square the values, keep the same probabilities.
- 5Var(X)=E(X2)−[E(X)]2=9.3−8.41Note E(X2)=[E(X)]2.
- 6=0.89, so the standard deviation is 0.943.Positive, as any variance must be.
Example 2 — finding an unknown constant
X takes the values 1,2,3 with P(X=x)=kx. Find k and E(X).
- 1The probabilities must sum to 1.This is always the equation that determines k.
- 2k(1)+k(2)+k(3)=1
- 36k=1, so k=61.The distribution is 61,62,63.
- 4E(X)=1(61)+2(62)+3(63)
- 5=61+4+9=614=37About 2.33 — skewed towards the larger values, as the increasing probabilities require.