Assume that $Y$ has a binomial distribution with parameters $p$ and $N$, where $N$ has a Poisson distribution with parameter $\lambda$. Find the marginal distribution for $Y$.
66. [10] Assume that a hen lays $N$ eggs, where $N$ has a Poisson distribution with parameter $\lambda$. Suppose each egg hatches with probability $p$ independently of the other eggs. Let $Y$ be the number of chicks that hatch. We can write $Y$ as the random sum
$$
Y=X_1+\cdots+X_N
$$
where $X_1, X_2, \ldots$ are independent Bernoulli random variables with parameter $p$.
61. [TM25] Consider Example 3.2.11. Find a Coxian distribution to match the given $\mathrm{H}_2$ distribution by first finding the Laplace-Stieltjes transform of the given distribution. Then find the two-stage Coxian distribution with the same Laplace-Stieltjes transform.
62. [HM18] Use convolution to show that if $X$ and $Y$ are independent exponential random variables with parameters $\alpha$ and $\beta$, respectively, where $\alpha \neq \beta$, then the density function $f$ of their sum $X+Y$ is given by
$$
f_{X+Y}(t)=\frac{\alpha \beta}{\alpha-\beta}\left(e^{-\beta t}-e^{-\alpha t}\right), \quad t \geq 0 .
$$
[We showed this, using the Laplace-Stieltjes transform, in Example 3.4.1.]
63. [10] Let $X$ be an Erlang- $k$ random variable with parameter $\mu$. Show that
$$
X^*[\theta]=\left(\frac{k \mu}{k \mu+\theta}\right)^k .
$$
Hint: Use the fact that the Laplace-Stieltjes transform of an exponential random variable with parameter $\alpha$ is
$$
\frac{\alpha}{\theta+\alpha}
$$
as well as the fact that $X$ can be represented as shown in Figure 3.2.5.
64. [HM15] Suppose $X$ is a gamma random variable with parameters $\beta$ and $\alpha$. Show directly from the definition that
$$
X^*[\theta]=\left(\frac{\alpha}{\theta+\alpha}\right)^\beta
$$
65. [M15] Assume that $Y$ has a binomial distribution with parameters $p$ and $N$, where $N$ has a Poisson distribution with parameter $\lambda$. Find the marginal distribution for $Y$.