Probability and Stochastic Homework 4 (Papoulis Chapter 5)

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).