##### Personal tools
•
You are here: Home Homework Solutions

# Homework Solutions

## Utah State University ECE 6010 Stochastic Processes Homework # 6 Solutions

1. Suppose is a sequence of independent r.v.s each of which is uniformly distributed on the interval . Define a sequence of r.v.s by , where . Show that converges in distribution to an exponential r.v. with p.d.f.

Here,

Now,

therefore,

we have , so

Therefore,

So it converges in distribution.

2. Suppose (i.p.) and that there is a constant such that for all . Show that (m.s.)

We have (i.p.) and . Therefore,

Define,

and

and let and be the corresponding indicator functions, so that .

Taking limit we have . Therefore we have,

Therefore, (i.p.) (m.s.) if .

3. Suppose (in distribution), where is a constant. Show that (i.p.)

(i.p.) .

 (by convergence in distribution)

Therefore,