Let X and Y are independent continuous real-valued random variables with pdf fX, fY respectively.
statistics
Description
1. Let X and Y are independent continuous real-valued random variables with pdf fX, fY respectively.
(a) Show that E[X] = ∫ ∞ 0 (1 − FX(x))dx − ∫ 0 −∞ FX(x)dx.
(b) Let Z = X + Y . Show that fZ(z) = ∫ ∞ −∞ fY (z − x)fX(x)dx. [Remark: This is known as the convolution formula.]
2. Let X and Y are independent Poisson random variables with parameter λ and µ respectively.
(a) Show that X + Y is a Poisson random variable with parameter λ + µ.
(b) What is the conditional probability P(X = x|X + Y = n). Show your steps.
(c) Hence, are X and X + Y independent?
3. (a) Let X be a continuous random variable with the pdf fX, and let Y = X2 . By considering the CDF of Y , express the fY (y) in terms of fX.
(b) In general when the transformation Y = g(X) is not one-to-one in the entire support of X, we cannot directly apply the Jacobian transformation. But suppose we can partition the support of X into two (or more) sets A1, A2, ..., Ak such that the transformation is one-to-one within each set, (like the set {X > 0} and {X < 0} in part
a) i.e. there exist some functions g1, g2, ..., gk such that Y = gi(X) when X ∈ Ai , i = 1, 2, ..., k and gi is one-to-one, then we can
