Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,
P(X = 1, Y = 0) = .3, P(X = 2, Y = 0) = .2 P(X = 0, Y = 1) = .2,
P(X = 1, Y = 1) = .2, P(X = 2, Y = 1) = 0.
a. Determine E(X) and E(Y ).
b. Find Cov(X, Y )
c. Find Cov(2X + 3Y, Y ).
2. Let X be a uniform random variable over the interval (0, 1).
Let Y = X(1-X). a. Derive the pdf for Y . b. Check the pdf you
found in (a) is a pdf. c. Use the pdf you found in (a) to find the
mean of Y . d. Compute the mean of Y by using the distribution for
X. e. Use the pdf of Y to evaluate P(|x-1/2|<1/8). You cannot
use the pdf for X. f. Use...
Let X,,X, and X, be independent uniform random
variables on [0,1] Write Y = X, +X, and Z = X+ X. a.) Compute
E[X,X,X,. (5 points) b.) Compute Var(X). (5 points) c.) Compute and
draw a graph of the density function fr. (15 points)
Let X and Y be independent discrete random variables with the
following PDFs:
x
0
1
2
f(x)=P[X=x]
0.5
0.3
0.2
y
0
1
2
g(y)= P[Y=y]
0.65
0.25
0.1
(a) Show work to find the PDF h(w) = P[W=w] = (f*g)(w) (the
convolution) of W = X + Y
(b) Show work to find E[X], E[Y] and E[W] (note that E[W] =
E[X]+E[Y])
1. Let X and Y be independent U[0, 1] random variables, so that
the point (X, Y) is uniformly distributed in the unit square.
Let T = X + Y.
(a) Find P( 2Y < X ).
(b). Find the CDF F(t) of T (for all real numbers t).
HINT: For any number t, F(t) = P ( X <= t) is just the area
of a part of the unit square.
(c). Find the density f(t).
REMARK: For a...
Let X and Y be independent and uniformly distributed random
variables on [0, 1]. Find the cumulative distribution and
probability density function of Z = X + Y.
Let X and Y be uniformly distributed independent random
variables on [0, 1].
a) Compute the expected value E(XY ).
b) What is the probability density function fZ(z) of Z = XY
?
Hint: First compute the cumulative distribution function FZ(z) =
P(Z ≤ z) using a double integral, and then differentiate in z.
c) Use your answer to b) to compute E(Z). Compare it with your
answer to a).
Let X and Y be independent and identical uniform distribution on
[0, 1]. Let Z=min(X, Y). Find E[Y-Z].
Hint: condition on whether Y=Z or not. What is the probability
Y=Z?