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Exercise 1
Let π be a continuous random variable with PDF
5 π₯4 for 0 < π₯ β€ 2 π(π₯) =32 0 otherwise
and let π = π2.
1. Find the CDF of π .
2. Find the PDF of π .
3. Find πΌ[π].
Exercise 2
Suppose that the joint PDF of π and π is
{
15 π₯2 for 0 β€ π¦ β€ 1 β π₯2 and β 1 β€ π₯ β€ 1
π(π₯, π¦) =4
0 otherwise
Determine the marginal PDFs of π and π .
Exercise 3
Let π and π be continuous random variables with joint PDF
π(π₯, π¦) = {60πβ(2π₯+3π¦) otherwisefor π₯, π¦ β₯ 0
1. Are π and π independent?
2. Find πΌ[π|π 2].
3. Find π(π π).
Exercise 4
Let π and π be two continuous random variables with joint PDF
{π₯ + 32 π¦2 for 0 β€ π₯, π¦ β€ 1
0 otherwise
Find the MAP and the ML estimates of π given π = π¦.
Exercise 5
Find the VC-dimension of the set of the hyperplanes in a π-dimensional space.
Hint: consider the problem of binary classification in βπ.