PML-Homework 01 Solved

Exercise 1
Let 𝑋 be a continuous random variable with PDF

 5 𝑥4 for 0 < 𝑥 ≤ 2 𝑓(𝑥) =32 0 otherwise

and let 𝑌 = 𝑋2.

1.  Find the CDF of 𝑌 .

2.  Find the PDF of 𝑌 .

3.  Find 𝔼[𝑌].

Exercise 2
Suppose that the joint PDF of 𝑋 and 𝑌 is

{
                                                                                              15 𝑥2 for 0 ≤ 𝑦 ≤ 1 − 𝑥2 and − 1 ≤ 𝑥 ≤ 1

𝑓(𝑥, 𝑦) =4

                                                                                             0           otherwise

Determine the marginal PDFs of 𝑋 and 𝑌 .

Exercise 3
Let 𝑋 and 𝑌 be continuous random variables with joint PDF

                                                                                          𝑓(𝑥, 𝑦) = {60𝑒−(2𝑥+3𝑦) otherwisefor 𝑥, 𝑦 ≥ 0

1.  Are 𝑋 and 𝑌 independent?

2.  Find 𝔼[𝑌|𝑋 2].

3.  Find 𝑃(𝑋 𝑌).

Exercise 4
Let 𝑋 and 𝑌 be two continuous random variables with joint PDF

                                                                                                  {𝑥 + 32 𝑦2              for 0 ≤ 𝑥, 𝑦 ≤ 1

                                                                                                     0                 otherwise

Find the MAP and the ML estimates of 𝑋 given 𝑌 = 𝑦.

Exercise 5
Find the VC-dimension of the set of the hyperplanes in a 𝑑-dimensional space.

Hint: consider the problem of binary classification in ℝ𝑑.