Exercise 1
Let ๐ be a continuous random variable with PDF
5 ๐ฅ4 for 0 < ๐ฅ ≤ 2 ๐(๐ฅ) =32 0 otherwise
and let ๐ = ๐2.
1. Find the CDF of ๐ .
2. Find the PDF of ๐ .
3. Find ๐ผ[๐].
Exercise 2
Suppose that the joint PDF of ๐ and ๐ is
{
15 ๐ฅ2 for 0 ≤ ๐ฆ ≤ 1 − ๐ฅ2 and − 1 ≤ ๐ฅ ≤ 1
๐(๐ฅ, ๐ฆ) =4
0 otherwise
Determine the marginal PDFs of ๐ and ๐ .
Exercise 3
Let ๐ and ๐ be continuous random variables with joint PDF
๐(๐ฅ, ๐ฆ) = {60๐−(2๐ฅ+3๐ฆ) otherwisefor ๐ฅ, ๐ฆ ≥ 0
1. Are ๐ and ๐ independent?
2. Find ๐ผ[๐|๐ 2].
3. Find ๐(๐ ๐).
Exercise 4
Let ๐ and ๐ be two continuous random variables with joint PDF
{๐ฅ + 32 ๐ฆ2 for 0 ≤ ๐ฅ, ๐ฆ ≤ 1
0 otherwise
Find the MAP and the ML estimates of ๐ given ๐ = ๐ฆ.
Exercise 5
Find the VC-dimension of the set of the hyperplanes in a ๐-dimensional space.
Hint: consider the problem of binary classification in โ๐.
