Mixed Distributions - Expectation and Variance

Question

A bike has probability of breaking down $p$, on any given day. The repair cost of the bike, whenever it breaks down, is distributed as a Gamma random variable with shape $\alpha$ and rate $\lambda$.

1. What is the expected repair cost, given that the bike broke down?
2. What is the expected annual repair cost?
3. What is the variance of the annual repair cost?

First, I want to define the event that the bike breaks down on a given day to be $X \sim Bern(p)$, and the repair cost (given that the bike broke down) to be $Y|X \sim Gamma(\alpha, \lambda)$.

1. $\mathbb{E}[Y\mid X] = \frac{\alpha}{\lambda}$

2. $\mathbb{E}[365X \cdot \mathbb{E}[Y\mid X]] = \frac{\alpha}{\lambda} \mathbb{E}[365X] = 365 p \frac{\alpha}{\lambda}$

3. $Var(365X \cdot \mathbb{E}[Y\mid X]) = 365^2 \cdot \mathbb{E}^2[Y\mid X] \cdot Var(X) = 365^2 \cdot \frac{\alpha^2}{\lambda^2} p (1-p)$

Am I on the right track here? I am having issues understanding if I used the notation correctly, or whether I even defined my random variables correctly.

Answer 1

Let $X$ be the number of breakdowns per year (A Poisson distribution with $\lambda=365p$) and $Y_i$ the cost of the $i$th breakdown (i.i.d Gamma distributions with shape α and rate λ) and S = $\sum_{i=1}^XY_i$ (a sum of a random number of i.i.d random variables).

1. As you say $\mathbb E(Y_i)=\frac{\alpha}{\lambda}$

For the next 2 we are working with $S$ and we can use these results. So,

2. $$\begin{align}\mathbb E(S)&=\mathbb E(X)\mathbb E(Y)\\&=365p\frac{\alpha}{\lambda}\end{align}$$.
3. $$\begin{align}Var(S)&=\mathbb E(X)Var(Y)+\mathbb E(Y)^2Var(X)\\ &=365p\frac{\alpha}{\lambda^2}+\left(\frac{\alpha}{\lambda}\right)^2365p\\ &=365p\frac{\alpha(1+\alpha)}{\lambda^2} \end{align}$$