Linear combinations of random variables notes

1Linear combinations of random variables

Syllabus 6.2

\(\text{E}(aX + bY) = a\text{E}(X) + b\text{E}(Y)\) always. For independent \(X\) and \(Y\), \(\text{Var}(aX + bY) = a^{2}\text{Var}(X) + b^{2}\text{Var}(Y)\), so variances ADD even for a difference. Constants shift the mean but not the variance. Combinations of independent normals are normal; sums of independent Poissons are Poisson.

Fig. 1: Ann's time A ~ N(25, 9) and Ben's time B ~ N(23, 16). The difference A − B ~ N(2, 25); the shaded area P(A − B < 0) = 0.3446 is the probability that Ann is faster

Worked example: Mean and variance of a transformed variable

\(\text{E}(X) = 12\) and \(\text{Var}(X) = 4\). Find the mean and variance of \(Y = 3X - 5\).

\(\text{E}(Y) = 3 \times 12 - 5 = 31\).
\(\text{Var}(Y) = 3^{2} \times 4 = 36\). The \(-5\) has no effect on the spread.

Answer: \(\text{E}(Y) = 31\), \(\text{Var}(Y) = 36\)

Write the combination, then its mean and variance.

Define the combination, e.g. \(D = X_1 + X_2 + X_3 - 2Y\), so the question becomes \(\text{P}(D > 0)\).
Mean: apply the coefficients directly.
Variance: square every coefficient and ADD, for sums and differences alike.
Separate items or one item scaled? \(n\) items: \(n\sigma^{2}\); one item times \(n\): \(n^{2}\sigma^{2}\).
Shape: normal stays normal; Poisson sums stay Poisson.
Standardise with the square root of the variance.

Watch out!

\(\text{Var}(3X - 5)\) is not \(3 \times 4 - 5\), and not \(3 \times 4\). The factor is squared and the constant disappears: \(9 \times 4 = 36\).

The commonest Paper 6 error: \(\text{Var}(A - B) = 9 - 16\). A variance can never be negative; for independent variables they always add.

Before adding Poisson parameters, put them on the same interval. A rate of 2 per hour and 0.5 per 10 minutes give \(\text{Po}(2 + 3) = \text{Po}(5)\) for an hour, not \(\text{Po}(2.5)\).

Revisit it: Linear combinations of random variables lesson · worked practice