Question

Consider two random variables X and Y . X can take the values 0, 1
and 2 andY cantakethevalues0and1. Youaretoldthat:
P(X=0)=0.4,P(X=1)=0.2, P(Y =0|X=0)=0.4,P(Y =0|X=1)=0.2,P(Y
=0|X=2)=0.3. Calculate P(Y =1),E(Y),Cov(X,Y),P(X=0|Y =0),P(Y =1|X+Y
≤2)

Answer #1

Consider the following bivariate distribution p(x, y) of two
discrete random variables X and Y.
Y\X
-2
-1
0
1
2
0
0.01
0.02
0.03
0.10
0.10
1
0.05
0.10
0.05
0.07
0.20
2
0.10
0.05
0.03
0.05
0.04
a) Compute the marginal distributions p(x) and p(y)
b) The conditional distributions P(X = x | Y = 1)
c) Are these random variables independent?
d) Find E[XY]
e) Find Cov(X, Y) and Corr(X, Y)

Let x and Y be two discrete random variables, where x Takes
values 3 and 4 and Y takes the values 2 and 5. Let furthermore the
following probabilities be given:
P(X=3 ∩ Y=2)= P(3,2)=0.3,
P(X=3 ∩ Y=5)= P(3,5)=0.1,
P(X=4 ∩ Y=2)= P(4,2)=0.4 and
P(X=4∩ Y=5)= P(4,5)=0.2.
Compute the correlation between X and Y.

Let x and Y be two discrete random variables, where x Takes
values 3 and 4 and Y takes the values 2 and 5. Let furthermore the
following probabilities be given:
P(X=3 ∩ Y=2)= P(3,2)=0.3,
P(X=3 ∩ Y=5)= P(3,5)=0.1,
P(X=4 ∩ Y=2)= P(4,2)=0.4 and
P(X=4∩ Y=5)= P(4,5)=0.2.
Compute the correlation between X and Y.

Consider two random variables X and Y such that E(X)=E(Y)=120,
Var(X)=14, Var(Y)=11, Cov(X,Y)=0.
Compute an upper bound to
P(|X−Y|>16)

. Let X and Y be two discrete random variables. The range of X
is {0, 1, 2}, while the range of Y is {1, 2, 3}. Their joint
probability mass function P(X,Y) is given in the table below:
X\Y 1
2
3
0
0
.25 0
1
.25
0
.25
2
0
.25 0
Compute E[X], V[X], E[Y], V[Y], and Cov(X, Y).

Consider two random variables, X and Y, with joint PDF
fxy(x,y)=e-2|y-x^2|-x x>=0
, y can be any value
fxy(x,y)=0 otherwise
(1) Determine fY|X(y|x)
(2)Determine E[Y|X=x]

The random variable X can take on the values 1, 2 and 3
and the random variable Y can take on the values 1, 3, and 4. The
joint probability distribution of X and Y is given in the following
table:
Y
1
3
4
X
1
0.1
0.15
0.1
2
0.1
0.1
0.1
3
0.1
0.2
a. What value should go in the blank cell?
b. Describe in words and notation the event
that has probability 0.2 in...

Problems
1. Two independent random variables X and Y
have the probability distributions as follows:
X 1 2 5
P (X) 0.2 0.5 0.3
Y 2 4
P (Y) 0.7 0.3
a) Let T = X + Y. Find all possible values of T.
Compute μ and . T σ T
b) Let U = X - Y. Find all possible values of U.
Compute μ U and σ U .
c) Show that μ T
= μ X +...

Consider the following joint distribution between random
variables X and Y:
Y=0
Y=1
Y=2
X=0
P(X=0, Y=0) = 5/20
P(X=0, Y=1) =3/20
P(X=0, Y=2) = 1/20
X=1
P(X=1, Y=0) = 3/20
P(X=1, Y=1) = 4/20
P(X=1, Y=2) = 4/20
Further, E[X] = 0.55, E[Y] = 0.85, Var[X] = 0.2475 and Var[Y] =
0.6275.
a. (6 points) Find the covariance between X and Y.
b. (6 points) Find E[X | Y = 0].
c. (6 points) Are X and Y independent?...

Let X and Y be independent discrete random variables with
pmf’s:
x
1
2
3
y
2
4
6
p(x)
0.2
0.2
0.6
p(y)
0.3
0.1
0.6
What is the probability that X + Y = 7

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