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 opalinus 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, y). as always, remember to state the range of x and y. then compute the joint probability mass function when (x, y) = (2,1). [hint: recall the formula for a conditional probability and note that we know the conditional probability and one of the marginal distributions].

(b) calculate the mean and variance of y. [hint: use the law of iterated expectations and variances].

(c) write down the formula for the joint (cumulative) distribution function of (x, y) using your result in part (a). as always, remember to state the relevant ranges of x and y. then compute the joint distribution function when x, y = (2,1). [hint: when writing down the joint distribution function, recall that the value of y must always be less than or equal to the value of x. if you are stuck on the second part, try making a table of the joint pmf of (x, y) up to the values x, y = (2,1)].

(d) write down the formula for the marginal probability mass function of y. as always, remember to state the range of y.

(e) note that y is less than x with zero probability so that, for example, p(x = 0, y = 1) = 0. use this and the fact that p (y=1) ≥ p(x=2, y=1) to determine whether x and y are independent. carefully show your argument.