As an acturial student, I have to do :
Show that for any two random variables V, W one has
Var(W) = E(Var(W|V )) + Var(E(W|V )).
Use the formula to solve the following problem: In a tourist office trips are organized at days with sunny weather only. Assume that on each day the weather is sunny with probability 0.7, independent of the other days. If a new guide starts to work on a sunny day, and if this guide faces N consecutive sunny days, the total number of customers until the first non-sunny day is Poisson distributed with parameter 5N, what is the expectation and variance of total customers that this guide faces until the first non-sunny day?
Is it right to use a geometric distribution for the sunny or non-sunny day ? If not which one do I need to use ?
For the first part of the total variance equation, take $$ Var(W) = E[W^2] - E^2[W], $$ by using the law of total expectation $$ Var(W) = E[E[W^2|X]] - (E[E[W|X]])^2= E[Var(W|X) + E^2[W|X]] - (E[E[W|X]])^2, $$ slightly rearranging the equation you'll get the formula \begin{align} Var(W) = & E[Var(W|X)] + E( E^2[W|X]) - (E[E[W|X]])^2\\ = & E[Var(W|X)] + Var[E(W|X)] \, . \end{align}
For the second part, denote $N$ the number of consecutive sunny day, that is $N \sim Geo(0.3)$ and the number of costumers $X$ each day $i$ is $Poisson(5)$, thus $$ E(\sum_{i=1}^NX_i) =EX_i EN=\frac{5}{0.3}. $$ For the variance, use the law of total variance that you proved in the first part.