1
$\begingroup$

$Y\mid X\sim\mathrm{Poisson}(X)$ and $X\sim\mathrm{uniform}(a,b)$

Find the marginal distribution, mean and variance of $Y$.

Then, what happens if $X$ is normal?

  • 1
    Hint for the "Then,..." part: In the definition of a Poisson random variable, are there any restrictions on the value of the parameter (usually denoted by $\lambda$)?2012-12-25

1 Answers 1

3

Assume that $0 < a < b$. Conditioned on the value of $X$, $Y$ is a Poisson random variable with conditional mean $E[Y\mid X] = X$ and conditional variance $\text{var}(Y\mid X) = X$, that is, $E[Y\mid X] = X ~\Rightarrow ~ E[Y] = E[E[Y\mid X]] = E[X] = \frac{a+b}{2}.$ Similarly, $E[Y^2] = E[E[Y^2\mid X]] = E[X^2 + X]$ which can also be computed from the distribution of $X$ and used in obtaining the variance of $Y$ via $\text{var}(Y) = E[Y^2]-(E[Y])^2.$ Alternatively, the variance formula (variance of a random variable is the sum of the mean of the conditional variance and the variance of the conditional mean) gives $\begin{align*} \text{var}(Y) &= E[\text{var}(Y\mid X)] + \text{var}(E[Y\mid X])\\ &= E[X] + \text{var}(X) \end{align*}$ and so all we need to know/compute further is the variance of the uniformly distributed random variable $X$.