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.
