[SOLVED] PML Homework 01

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 ℝ^𝑑.