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
extend the result as fY (y) = ∑
k
i=1
fX(g
−1
i
(y))|Ji
| where Ji
is the corresponding Jacobian of the
transformation gi
.
Let X1, X2
i.i.d. ∼ N (0, σ2
).
i. Find the joint pdf of Y1 := X
2
1 + X
2
2
, Y2 := √
X1
X2
1 + X2
2
.
ii. Are Y1 and Y2 independent?
4. (a) For the hierarchical model
Y |Λ ∼ Poisson(Λ) and Λ ∼ Gamma(α, β),
find the marginal distribution, mean, and variance of Y . Show that the marginal distribution
of Y is a negative binomial if α is an integer.
(b) Show that the three-stage model
Y |N ∼ Binomial(N, p), N|Λ ∼ Poisson(Λ), and Λ ∼ Gamma(α, β)
leads to the same marginal distribution of Y .
5. Suppose the distribution of Y , conditional on X = x, is N(x, x2
) and that the marginal distribution
of X is Uniform(0, 1).
(a) Find E[Y ], V ar(Y ) and Cov(X, Y ).
(b) Prove that Y/X and X are independent.
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