Question

(a) Given two independent uniform random variables X, Y in the interval (−1, 1), find E |X − Y |.

(b) Let X, Y be as in (a). Find the support and density of the random variable Z = |X − Y |.

(c) From (b), compute the mean of Z and check whether you get the same answer as in (a)

Answer #1

a)

f(x,y) = 1

= 1/3

b)

c)

Let X and Y be independent random variables, with X following
uniform distribution in the interval (0, 1) and Y has an Exp (1)
distribution.
a) Determine the joint distribution of Z = X + Y and Y.
b) Determine the marginal distribution of Z.
c) Can we say that Z and Y are independent? Good

STAT 180 Let X and Y be independent exponential random variables
with mean equals to 4.
1) What is the covariance between XY and X.
2) Let Z = max ( X, Y). Find the Probability Density Function
(PDF) of Z.
3) Use the answer in part 2 to compute the E(Z).

Let X and Y be independent random variables each having the
uniform distribution on [0, 1].
(1)Find the conditional densities of X and Y given that X > Y
.
(2)Find E(X|X>Y) and E(Y|X>Y) .

Let U1 and U2 be independent Uniform(0, 1) random variables and
let Y = U1U2.
(a) Write down the joint pdf of U1 and U2.
(b) Find the cdf of Y by obtaining an expression for FY (y) =
P(Y ≤ y) = P(U1U2 ≤ y) for all y.
(c) Find the pdf of Y by taking the derivative of FY (y) with
respect to y
(d) Let X = U2 and find the joint pdf of the rv pair...

Let X and Y be independent random variables, uniformly
distribued on the interval [0, 2]. Find E[e^(X+Y) ].

Let X and Y be independent exponential random variables with
respective parameters 2 and 3.
a). Find the cdf and density of Z = X/Y .
b). Compute P(X < Y ).
c). Find the cdf and density of W = min{X,Y }.

Let X and Y be independent random variables with density functions given by fX (x) = 1/2, −1 ≤ x ≤ 1 and fY (y) = 1/2, 3 ≤ y ≤ 5. Find the density function of X-Y.

Let X and Y be two independent random variables with
μX =E(X)=2,σX =SD(X)=1,μY =2,σY =SD(Y)=3.
Find the mean and variance of
(i) 3X
(ii) 6Y
(iii) X − Y

Let X and Y be two independent random variables. Given the
marginal pdfs indicated below, find the cdf of Y/X. (Hint: Consider
two cases, 0 ≤ w ≤ 1 and 1.) (a) fx (x) =1, 0 ≤ x ≤ 1, and fγ
(y)=1, 0 ≤ y ≤ 1 (b) fx (x)=2x,0 ≤x ≤1, and fy(y)=2y, 0 ≤y ≤1

Let X and Y be two continuous random variables with joint
probability density function
f(x,y) =
6x 0<y<1, 0<x<y,
0 otherwise.
a) Find the marginal density of Y .
b) Are X and Y independent?
c) Find the conditional density of X given Y = 1 /2

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