Given a fair coin and a fair 4 sided die, Jonathan and Joseph run the following experiment. Theyroll the die (letYcorrespond to the random variable associated with the roll). If they roll aytheythe flip the coinynumber of times, letZcorrespond to the random variable associated with thetotal number of heads they flip.(i) Find the marginal PMFs ofYandZ(i. e. findPY(y) andPZ(z)).(ii) Find the conditional PMFs ofYandZ(i. e. findPY|Z(y|z) andPZ|Y(z|y))

the battle of gettysburg is considered the "high water mark" of the confederacy because it marked the greatest advance of the confederate forces against the union.

is there supposed to be a diagram or a graph for this one? ?

step-by-step explanation:

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: