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 âˆ’ 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) â‰¥ P(X=2, Y=1) to determine whether X and Y are independent.

Carefully show your argument.