5. [24 points] Let X be the total number of individuals of an endangered lizard species that are observed in a region on a given day. This observed number is assumed to be distributed according to a Poisson distribution with a mean of 3 lizards. The endangered lizards can belong to either of two sub-species: graham or opalinus. Let Y be the number of graham lizards observed during this study [note that the total observed number of any lizard is denoted by X, so the observed number of opaii'nus lizards is given by X Y). It is known that graham lizards are by far the most common. In particular, the conditional distribution of the number of graham lizards {Y} given the total number of all lizards (X) is Binomial with parameters n = X and p = 0.8. (a) Write down the joint probability mass function of (X, i"). As always, remember to state the range ofx and 3!. Then compute the joint probability mass function when {any} = (2,1). [Hintz recall the formula for a conditional probability and note that we know the conditional probability and one of the marginal distributions]. (10 pts) (b) Calculate the mean and variance of Y. [Hintz use the law of iterated expectations and variances]. (10 pts) (c) Write down the formula for the marginal probability mass function of Y. As always, remember to state the range of y. (4 pts}