Question

Let LaTeX: X,YX , Y be two discrete random variables that have the following joint distribution:

x = 0 1

y = -1 0.18 0.12

0 ? 0.20

1 0.12 0.08

(a) Determine the following probabilities:

LaTeX: P(X=0, Y=0) P ( X = 0 , Y = 0 ), LaTeX: P(X\le 0,Y\le 0)P ( X ≤ 0 , Y ≤ 0 )

(b) Find the marginal distribution of LaTeX: YY.

(c) What is the conditional distribution of LaTeX: XX given LaTeX: Y=-1Y = − 1?

(d) Are LaTeX: X,YX , Y independent? State your reason clearly.

(e) What is LaTeX: E\left(XY\right)E ( X Y )?

Answer #1

The joint distribution is given to be as follows,

(with X on the horizontal and Y on the vertical). Here the missing value is obtained by subtracting the sum of all other values from 1 since the total probability must be equal to 1. Then,

a)

b) The marginal distribution of Y is obtained as,

c) Conditional distribution of X given Y = -1,

d) Expected value of XY.

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, Y be two random variables with a joint pmf
f(x,y)=(x+y)/12 x=1,2 and y=1,2
zero elsewhere
a)Are X and Y discrete or continuous random variables?
b)Construct and joint probability distribution table by writing
these probabilities in a rectangular array, recording each marginal
pmf in the "margins"
c)Determine if X and Y are Independent variables
d)Find P(X>Y)
e)Compute E(X), E(Y), E(X^2) and E(XY)
f)Compute var(X)
g) Compute cov(X,Y)

For the discrete joint distribution below, find E[X|Y = 1].
Y
0 1
X 1 0.32' 0.15
2 0.08 0.12
3 0.10 0.23

Suppose that you have two discrete random variables X and Y with
the following joint probability distribution, which is similar to
the example in class. Fill in the marginal probabilities below.
Possible Values of X
Possible
Values
of Y
1
2
3
4
1
0
18
18
14
2
18
14
18
0
Please input the exact answer in either decimal or fraction
form.

Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ? + 2)/(B + 2) for 0 ≤ X < 3, 0 ≤ Y < 3
Where B=2.
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 2

Let X and Y be discrete random variables, their joint pmf is
given as ?(x,y)= ?(? + ? − 2)/(B + 1) for 1 < X ≤ 4, 1 < Y ≤
4 Where B is the last digit of your registration number ( B=3) a)
Find the value of ? b) Find the marginal pmf of ? and ? c) Find
conditional pmf of ? given ? = 3

SOLUTION REQUIRED WITH COMPLETE STEPS
Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ? + 1)/(B + 1) for 0 ≤ X < 3, 0 ≤ Y < 3
(Where B=7)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 1

SOLUTION REQUIRED WITH COMPLETE STEPS
Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ?)/(B + 1) for 0 < X ≤ 3, 0 < Y ≤ 3
(Where B=5)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 2

. 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).

The joint probability distribution of two random variables X and
Y is given in the following table
X Y →
↓
0
1
2
3
f(x)
2
1/12
1/12
1/12
1/12
3
1/12
1/6
1/12
0
4
1/12
1/12
0
1/6
f(y)
a) Find the marginal density of X and the marginal density of Y.
(add them to the above table)
b) Are X and Y independent?
c) Compute the P{Y>1| X>2}
d) Compute the expected value of X.
e)...

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