AHL 4.14 (HL)—Random variable transformations

A transformation maps a random variable to a new variable, such as Y=aX+b. For expectation and variance, E(Y)=aE(X)+b and Var(Y)=a²Var(X); a shift changes location while a scale changes spread.

For a discrete variable, transform each possible value and keep its probability. For a continuous variable, monotonic changes also alter the density and interval interpretation, so do not transform only the mean and ignore the distribution.

If E(X)=10 and Var(X)=4, then Y=3X−2 has E(Y)=28 and Var(Y)=36. The subtraction moves every value down by two; it does not reduce the variance.

Variance is squared in the scale factor and is not measured in X's units. Check whether the transformation reverses order or changes the support before interpreting probabilities.

For $Y=\sum a_iX_i+b$, $E(Y)=\sum a_iE(X_i)+b$; if the $X_i$ are independent, $Var(Y)=\sum a_i^2Var(X_i)$. The sample mean $\bar X$ is an unbiased estimator of $\mu$, and $s_{n-1}^2=\sum f_i(x_i-\bar x)^2/(n-1)$ is an unbiased estimator of $\sigma^2$. Independence is required for adding variances without covariance terms.