Question

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)

Answer #1

The random variables, X and Y , have the joint pmf
f(x,y)=c(x+2y), x=1,2 y=1,2 and zero otherwise.
1. Find the constant, c, such that f(x,y) is a valid pmf.
2. Find the marginal distributions for X and Y .
3. Find the marginal means for both random variables.
4. Find the marginal variances for both random variables.
5. Find the correlation of X and Y .
6. Are the two variables independent? Justify.

Suppose that X and Y are two jointly continuous random variables
with joint PDF
??,(?, ?) =
??
??? 0 ≤ ? ≤ 1 ??? 0 ≤ ? ≤ √?
0
??ℎ??????
Compute and plot ??(?) and ??(?)
Are X and Y independent?
Compute and plot ??(?) and ???(?)
Compute E(X), Var(X), E(Y), Var(Y), Cov(X,Y), and
Cor.(X,Y)

Suppose X and Y are continuous random variables with joint
density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a). Compute the joint CDF F(x,y).
(b). Compute the marginal density for X and Y .
(c). Compute Cov(X,Y ). Are X and Y independent?

9. Suppose X and Y are continuous random variables with joint
density function f(x,y) = x + y for 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1.
(a). Compute the joint CDF F(x,y).
(b). Compute the marginal density for X and Y .
(c). Compute Cov(X,Y ). Are X and Y independent?

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

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 continuous random variables with joint
distribution function F(x, y), and let g(X, Y ) and h(X, Y ) be
functions of X and Y . Prove the following:
(a) E[cg(X, Y )] = cE[g(X, Y )].
(b) E[g(X, Y ) + h(X, Y )] = E[g(X, Y )] + E[h(X, Y )].
(c) V ar(a + X) = V ar(X).
(d) V ar(aX) = a 2V ar(X).
(e) V ar(aX + bY ) = a...

Problems 9 and 10 refer to the discrete random variables X and Y
whose joint distribution is given in the following table.
Y=-1
Y=0
Y=1
X=1
1/4
1/8
0
X=2
1/16
1/16
1/8
X=3
1/16
1/16
1/4
P9: Compute the marginal distributions of X and Y, and use these
to compute E(X), E(Y), Var(X), and Var(Y).
P10: Compute Cov(X, Y) and the correlation ρ for the random
variables X and Y. Are X and Y independent?

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

SOLUTION REQUIRED WITH COMPLETE STEPS
Let X and Y be discrete random variables, their joint pmf is
given as Px,y = ?(? + ? − 2)/(B + 1) for 1 < X ≤ 4, 1 < Y ≤ 4
(Where B=2)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 3

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