Let X and Y be continuous random variables. Define a new random variable Z according to the following process. First flip a biased coin with probability α of obtaining a head. If the result is heads, sample from X; otherwise sample from Y . Such a distribution is called a mixture of X and Y.
a) Derive a formula for the cdf of Z in terms of α and the cdfs of X and Y.
b) Derive a formula for the pdf of Z in terms of α and the pdfs of X and Y . c) Let W ∼ 0.3 · Norm(8, 2) : 0.7 · Norm(16, 3) (that is, a 30-70 mix of two normal distributions). What is P(W ≤ 12)?