Exercise: 6
1. A fair die is rolled at random three times, Let the random variable Xi be equal to the no. that
appear on the ith trial for i=1,2,3. Let Y=max(X1, X2, X3). Find distribution function and
probability density function of Y.
2. Show that correlation coefficient ρ(X,Y), where X and Y are two random variable is
Var ( X ) + Var (Y ) − Var ( X − Y )
.
=
2 Var ( X ).Var (Y )
3. X and Y have the joint probability density function f(x, y) = 1, -x<y<x, 0<x<1.
0, elsewhere.
Draw the graph of E(Y/X=x) as a function of x and also that of E(X/Y=y) as a function of y.
4. Suppose the conditional probability density function of X2, given X1=x1>0 is
( x1 + x 2 ) − x2
f(X2/X1=x)(x2/X1) =
e , x2 > 0
( x1 + 1)
= 0, Otherwise.
( x + 1) − x1
And that f X 1 ( x1 ) = 1
e , x1 > 0
2
= 0,
Otherwise.
Find E(X1/X2=x2), E(X1) and Correlation (X1, X2).
5. Let (X1, X2) be a two- dimensional discrete random variables with joint probability function
Px1 , x2 ( X 1 , X 2 )
X2
X1
1
2
3
4
(i) Compute Correlation (X1, X2)
1
1/8
1/1
1/8
3/16
2
1/16
1/16
1/8
1/4
(ii) Are X1 and X2 independent random variables?
Exercise: 6
6. Using the result E(X, Y) = E(XE(Y/X)), show that Cov(X,E(Y/X)) = Cov(X,Y).
7. Consider the joint probability density function of (X, Y) given by
fXY (x,y) = 2-x-y
0<x<1, 0<y<1
= 0,
Otherwise.
(i) Find the conditional probability density function of Y given X=x.
(ii) Compute E(Y/X= x), E(Y)
(iii) Verify that E(Y) = E(E(Y/X = x)).
8. Let X, Y, Z be random variables and a, b be constants. Prove that
(i) Cov(X, aY + b) = a Cov (X, Y)
(ii) ρ(X, aY + b) = ρ(X,Y) for a > 0
9. Let X1, X2, X3 be three independent random variables each with variance σ2. If we define
new random variables
3 −1
3− 3
W3 = ( 2 − 1) X 2 + (2 − 2 ) X 3 ,
W1=X1, W2 =
X1 +
X2,
2
2
show that ρ(W1,W2) = ρ(W2,W3) = 1/2 while W1 and W3 are uncorrelated.
10. A Fair die is successively rolled. Let X and Y denote respectively, the numbers of rolls
necessary to obtain a 6 and a 5.
Find: (i) E(X) ; (ii) E[X/Y=1] ; (iii) E(X/Y =5).

Problem sheet 3

Assignment 1 FSV (MAL102)

Math 250A Formula Sheet Alternate Form of Dot Product u ⋅ v = u v