If x + 
, Find the value of x² + 
.
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6.1.6 Solved Problems
Problem
Let X,Y and Z be three jointly continuous random variables with joint PDF
fXYZ(x,y,z)={
1
3
(x+2y+3z) 0≤x,y,z≤1 0 otherwise
Find the joint PDF of X and Y, fXY(x,y).
Solution
Problem
Let X,Y and Z be three independent random variables with X∼N(μ,σ2), and Y,Z∼Uniform(0,2). We also know that
E[X2Y+XYZ]=13, E[XY2+ZX2]=14.
Find μ and σ.
Solution
Problem
Let X1, X2, and X3 be three i.i.d Bernoulli(p) random variables and
Y1=max(X1,X2), Y2=max(X1,X3), Y3=max(X2,X3), Y=Y1+Y2+Y3.
Find EY and Var(Y).
Solution
Problem
Let MX(s) be finite for s∈[−c,c], where c>0. Show that MGF of Y=aX+b is given by
MY(s)=esbMX(as),
and it is finite in [−
c
|a|
,
c
|a|
].
Solution
Problem
Let Z∼N(0,1) Find the MGF of Z. Extend your result to X∼N(μ,σ).
Solution
Problem
Let Y=X1+X2+X3+...+Xn, where Xi's are independent and Xi∼Poisson(λi). Find the distribution of Y.
Solution
Problem
Probability Generating Functions (PGFs): For many important discrete random variables, the range is a subset of {0,1,2,...}. For these random variables it is usually more useful to work with probability generating functions (PGF)s defined as
GX(z)=E[zX]=
∞
∑
n=0 P(X=n)zn,
for all z∈R that GX(z) is finite.
Show that GX(z) is always finite for |z|≤1.
Show that if X and Y are independent, then
GX+Y(z)=GX(z)GY(z).
Show that
1
k!
dkGX(z)
dzk
|z=0=P(X=k).
Show that
dkGX(z)
dzk
|z=1=E[X(X−1)(X−2)...(X−k+1)].
Solution
Problem
Let MX(s) be finite for s∈[−c,c] where c>0. Prove
limn→∞ [MX(
s
n
)]n=esEX.
Solution
Problem
Let MX(s) be finite for s∈[−c,c], where c>0. Assume EX=0, and Var(X)=1. Prove
limn→∞ [MX(
s
√
n
)]n=e
s2
2
.
Note: From this, we can prove the Central Limit Theorem (CLT) which is discussed in Section 7.1.
Solution
Problem
We can define MGF for jointly distributed random variables as well. For example, for two random variables (X,Y), the MGF is defined by
MXY(s,t)=E[esX+tY].
Similar to the MGF of a single random variable, the MGF of the joint distributions uniquely determines the joint distribution. Let X and Y be two jointly normal random variables with EX=μX, EY=μY, Var(X)=σ
2
X
, Var(Y)=σ
2
Y
, ρ(X,Y)=ρ . Find MXY(s,t).
Solution
Problem
Let X=[ X1 X2 ] be a normal random vector with the following mean vector and covariance matrix
m=[ 0 1 ],C=[ 1 −1 −1 2 ].
Let also
A=[ 1 2 2 1 1 1 ],b=[ 0 1 2 ],Y=[ Y1 Y2 Y3 ]=AX+b.
Find P(0≤X2≤1).
Find the expected value vector of Y, mY=EY.
Find the covariance matrix of Y, CY.
Find P(Y3≤4).
Solution
Problem
(Whitening/decorrelating transformation) Let X be an n-dimensional zero-mean random vector. Since CX is a real symmetric matrix, we conclude that it can be diagonalized. That is, there exists an n by n matrix Q such that
QQT=I(I is the identity matrix), CX=QDQT,
where D is a diagonal matrix
D=[ d11 0 ... 0 0 d22 ... 0 . . . . . . . . . . . . 0 0 ... dnn ].
Now suppose we define a new random vector Y as Y=QTX, thus
X=QY.
Show that Y has a diagonal covariance matrix, and conclude that components of Y are uncorrelated, i.e., Cov(Yi,Yj)=0 if i≠j.
Solution
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Introduction to Probability by Hossein Pishro-Nik is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License