Sums of random variables

Let $X$ and $Y$ be independent variables, $X$ being equally likely to take any value in $\left\{ 0,1, \ldots , m\right\}$, and $Y$ similarly in $\left\{ 0,1, \ldots , n\right\}$. Find the mass function of $Z = X+Y$. The random variable $Z$ is said to have the trapezoidal distribution.

Let $X$ and $Y$ have the joint mass function

Find the mass functions of $U = X+Y$ and $V = X-Y$.

Let $X$ and $Y$ be independent geometric random variables with respective parameters $\alpha$ and $\beta$. Show that

Let $\left\{ X_{r}: 1 \leq r \leq n\right\}$ be independent geometric random variables with parameter $p$. Show that $Z = \sum_{r = 1}^{n} X_{r}$ has a negative binomial distribution.

Sam rolls $6 n$ dice once; he needs at least $n$ sixes. Isaac rolls $6(n+1)$ dice; he needs at least $n+1$ sixes. Who is more likely to obtain the number of sixes he needs?

Let $N$ be Poisson distributed with parameter $\lambda$. Show that, for any function $g$ such that the expectations exist, $\mathbb {E}\left[N g(N)\right] = \lambda \mathbb {E}\left[g(N+1)\right]$. More generally, if $S = \sum_{r = 1}^{N} X_{r}$, where $\left\{ X_{r}: r \geq 0\right\}$ are independent identically distributed non-negative integer-valued random variables, show that

Let $S = \sum_{i = 1}^{N} X_{i}$, where the $X_{i}, i \geq 1$, are independent, identically distributed random variables with mean $\mu$ and variance $\sigma^{2}$, and $N$ is positive, integer-valued, and independent of the $X_{i}$. Show that $\mathbb {E}\left[S\right] = \mu \mathbb {E}\left[N\right]$, and

Let $X$ and $Y$ be independent random variables with the geometric distributions $f_{X}(k) = p q^{k-1}$, $f_{Y}(k) = \lambda \mu^{k-1}$, for $k \geq 1$, where $p+q = \lambda +\mu = 1$ and $q \neq \mu$. Write down $\mathbb {P}\left(X+Y = n+1, X = k\right)$, and hence find the distribution of $X+Y$, and the conditional distribution of $X$ given that $X+Y = n+1$. Does anything special occur when $q = \mu$?