Problems: 6.29, 6.30, 6.32, 6.33 6.34

Also:

1. Suppose $X$ and $Y$ are independent random variables, each uniformly distributed on $(0, 2)$. Find the probability density function of $Z = X + Y$ using the convolution of the densities of $X$ and $Y$. Check your answer with the example in Lecture 25.

2. Suppose $Z_1, Z_2, \ldots, Z_n$ are independent, standard normal random variables and let $W = Z_1^2 + Z_2^2 + \cdots + Z_n^2$. Explain why $W$ has a gamma distribution with parameters $\alpha = \frac{n}{2}$ and $\lambda = \frac{1}{2}$. That is, why $W$ has a chi-squared distribution with $n$ degrees of freedom

For problem set due 18 November: Suppose $X$ and $Y$ are independent binomial random variables, each with probability of success $p$. Suppose $X$ has $n$ trials and $Y$ has $m$ trials. Use the convolution of the probability mass functions to show that $X + Y$ is binomial with $n + m$ trials and probability of success $p$.