Probabilistic Machine Learning Homework 1

Description

Exercise 1
Let be a continuous random variable with PDF
and let .
1. Find the CDF of .
2. Find the PDF of .
3. Find .
𝑋
𝑓(𝑥) = {
5
32 𝑥4
0
for 0 < 𝑥 ≤ 2
otherwise
𝑌 = 𝑋2
𝑌
𝑌
𝔼[𝑌 ]
Exercise 2
Suppose that the joint PDF of and is
Determine the marginal PDFs of and .
𝑋 𝑌
𝑓(𝑥, 𝑦) = {
15
4 𝑥2
0
for 0 ≤ 𝑦 ≤ 1 − 𝑥 and − 1 ≤ 𝑥 ≤ 1 2
otherwise
𝑋 𝑌
Exercise 3
Let and be continuous random variables with joint PDF
1. Are and independent?
2. Find .
3. Find .
𝑋 𝑌
𝑓(𝑥, 𝑦) = {
6𝑒−(2𝑥+3𝑦)
0
for 𝑥, 𝑦 ≥ 0
otherwise
𝑋 𝑌
𝔼[𝑌 |𝑋 > 2]
𝑃(𝑋 > 𝑌 )
Exercise 4
Let and be two continuous random variables with joint PDF
Find the MAP and the ML estimates of given .
𝑋 𝑌
{
𝑥 + 3
2 𝑦2
0
for 0 ≤ 𝑥, 𝑦 ≤ 1
otherwise
𝑋 𝑌 = 𝑦
Exercise 5
Find the VC-dimension of the set of the hyperplanes in a -dimensional space.
Hint: consider the problem of binary classification in .
𝑑
ℝ𝑑