Suppose \(X \sim \text{Bin}(10,0.2)\) and \(Y \sim \text{Bin}(5,0.2).\)

(a) Say \(X_1,\ldots,X_{10}\) are iid with \(X = X_1 + \cdots + X_{10}\).  What distribution for the \(X_i\) makes this statement true?

(b) Say \(Y_1,\ldots,Y_{5}\) are iid with \(Y = Y_1 + \cdots + Y_{5}\).  What distribution for the \(Y_i\) makes this statement true?

(c) Write \(X + Y\) in terms of \(X_1,\ldots,X_{10}\) and \(Y_1,\ldots,Y_{5}\).

(d) What is the distribution of \(X + Y\)?

[Author Mark Huber. Licensed under Creative Commons.]