Probability and Stochastic
Homework 4
(Papoulis Chapter 5)
1. Suppose 𝑌 = 𝑥 2 − 3𝑥 + 1 and:
X
p
-2
1
5
-1
1
10
0
1
5
1
1
10
2
2
5
(a) Find 𝐸 (𝑌).
(b) Find 𝑉𝑎𝑟(𝑥 ).
2. The RV 𝑥 is a uniform in the interval (0, 1). Find the
density of the 𝑦 = − ln 𝑥.
[Problem 5-6 of Papoulis]
3. Prove: 𝜎 2 = 𝐸 (𝑥 2 ) − 𝜇2 .
4. For Bernoulli trial, prove: 𝜎 2 = 𝑛𝑝𝑞 (Hint: Use previous
question)
5. Show that, if the RV 𝑥 has a Cauchy density with 𝛼 = 1
and 𝑦 = tan−1 𝑥, then 𝑦 is uniform in the interval (
−𝜋 𝜋
2
, 2 ).
[Problem 5-11 of Papoulis]
6. Given that RV 𝑥 of continuous type, we form the RV 𝑦 =
𝑔(𝑥 ). Find 𝑓𝑦 (𝑦) if 𝑔(𝑥 ) = 2𝐹𝑥 (𝑥 ) + 4.
[Problem 5-14 of Papoulis]
7. We place at random 200 points in the interval (0, 100).
The distance from 0 to the first random point is an RV 𝑧.
Find 𝐹𝑧 (𝑧) exactly.
[Problem 5-7 of Papoulis]
8. We place at random 200 points in the interval (0, 100).
The distance from 0 to the first random point is an RV 𝑧.
Find 𝐹𝑧 (𝑧) using the Poisson approximation.
[Problem 5-7 of Papoulis]
9. The random variable 𝑥 is uniform in the interval
(−2𝜋, 2𝜋). Find 𝑓𝑦 (𝑦) if 𝑦 = 𝑥 3 .
[Problem 5-12 of Papoulis]
10. The random variable 𝑥 is uniform in the interval
(−2𝜋, 2𝜋). Find 𝑓𝑦 (𝑦) if 𝑦 = 2 sin(3𝑥 + 40°).
[Problem 5-12 of Papoulis]
11. For 𝑏(𝑛, 𝑝), determine Probability-generating function.
12. Determine Probability-generating function for Poisson
distribution (with 𝜆 parameter).