Geometric density Statistics/Probability

[nikki92]

Homework Statement

Suppose that X_1, X_2, ... are identical and independently distributed with F_x1(x) = exp(x)/(1+exp(x)) for -infinity < x < infinity

Suppose that N independent of X_i has geometric density f_N (n) = P(N=n) = p(1-p)^(n-1) for n =1,2,3,... and 0<p<1

Let Z = max{X_1, X_2,...}.

What is the cumulative distribution of F_Z(z)?

For n>0 where n is an integer and z is any real number, what is P(N=n | Z less than or equal to z)


How would I start this problem?

Homework Equations

The Attempt at a Solution

 

[Ray Vickson]

Homework Statement

Suppose that X_1, X_2, ... are identical and independently distributed with F_x1(x) = exp(x)/(1+exp(x)) for -infinity < x < infinity

Suppose that N independent of X_i has geometric density f_N (n) = P(N=n) = p(1-p)^(n-1) for n =1,2,3,... and 0<p<1

Let Z = max{X_1, X_2,...}.

What is the cumulative distribution of F_Z(z)?

For n>0 where n is an integer and z is any real number, what is P(N=n | Z less than or equal to z)

How would I start this problem?

Homework Equations

The Attempt at a Solution

I think your statement is incorrect: I would bet that
$$Z = \max \{ X_1, X_2, \ldots, X_N \},$$ so on the event {N = n} we have
$$Z = \max \{ X_1, X_2, \ldots, X_n \}.$$
So, to start, you need do get
$$P \{ Z \leq z | N = n \}$$
for n = 1,2, ... . First off, what is this when n = 1?

RGV