Question

Consider a succession of independent Bernoulli tests of parameter p. Let X be the number of failure before the first success and Y the number of failure between the first success and the second.

a) Calculate the joint density function of X and Y.

b) Calculate the conditional density function of X given Y = y.

c) Calculate the conditional density function of Y given X = x.

d) Are the variables X and Y independent? Argue your answer.

Answer 1

a)

The joint density function of X and Y is,

f(X = x, Y = y) = Probability of x failures and then a success and then y failures followed by a success

= (1-p)^x p * (1-p)^y * p = (1-p)^x+y p²

f(X = x, Y = y) = (1-p)^x+y p²   for x, y = 0, 1, 2, ...

b)

The PMF for Y is,

f(Y = y) = (1-p)^y p

The conditional density function of X given Y = y is f(X = x | Y = y)

= f(X = x, Y = y) / f(Y = y)

= (1-p)^x+y p² / (1-p)^y p

= (1-p)^xp

Thus,

f(X = x | Y = y) = (1-p)^xp

(c)

The PMF for X is,

f(X = x) = (1-p)^x p

The conditional density function of Y given X = x is  f(Y = y | X = x)

= f(Y = y, X = x) / f(X = x)

= (1-p)^x+y p² / (1-p)^x p

= (1-p)^y p

Thus,

f(Y = y | X = x) = (1-p)^yp

(d)

Since, (Y = y | X = x) = f(Y = y) and (X = x | Y = y) = f(Y = X), the variables X and Y are independent.