2 - People.stat.sfu.ca

4. Suppose X is Uniform on [0,1] and Y = sin(4πX). Find the density of
Y .
Solution: This transformation is many to 1. For each y ∈ (−1, 1) there
are 4 corresponding x values which map to y. The density of Y is
∑
1/|dy/dx|
x:sin(4πx)=y
We find dy/dx = 4π cos(x) and |dy/dx| = 4π √ 1 − sin 2 (x). Evaluated
at any x for which sin(4πx) = y this is 4π √ 1 − y 2 . This makes
1
f Y (y) = 4
4π √ 1 − y 1(−1 < y < 1) = 1
2 π √ 1(−1 < y < 1).
1 − y2 5. Suppose X and Y have joint density f(x, y). Prove from the definition
of density that the density of X is g(x) = ∫ f(x, y) dy.
Solution: I defined g to be the density of X provided
∫
P (X ∈ A) = g(x)dx
but
∫
A
∫
g(x)dx =
∫
=
A
∫
A×(−∞,∞)
A
f(x, y) dydx
f(x, y)dxdy
= P ((X, Y ) ∈ A × (−∞, ∞))
= P (X ∈ A)
Notice that I have used the fact that f is the density of (X, Y ) in the
middle of this.
6. Suppose X is Poisson(θ). After observing X a coin landing Heads with
probability p is tossed X times. Let Y be the number of Heads and Z
be the number of Tails. Find the joint and marginal distributions of Y
and Z.
4