6 The joint PMF of X and Y is given by
y\x
-1
0
1
-1
p
q
p
0
q
0
q
1
p
q
p
(a) Describe the possible values of p and q.
(b) Find the marginal PMFs of X and Y .
(c) Find the conditional PMF of Y given X = x for x = −1, 0,
1
(d) Find the conditional expectation of Y given X = x for x =
−1, 0, 1,...
let p ( x , y )be a joint pmf of Xand Y.
p ( x , y )
y = 0
y = 1
y = 2
x = 0
.12
.10
.08
x = 1
.13
.17
.10
x = 2
.15
.15
0
(a) Find the marginal pmf's of Xand Y.
(b) Determine whether Xand Yare independent.
c) Find Correlation (X,Y)
Let the joint pmf of X and Y be defined by f (x, y) = c(x + y),
x =0, 1, 2, y = 0, 1, with y ≤ x.
1. Are X and Y independent or dependent? Why or why not?
2. Find g(x | y) and draw a figure depicting the conditional
pmfs for y =0 and 1.
3. Find h(y | x) and draw a figure depicting the conditional
pmfs for x = 0, 1 and2.
4....
Let the joint pmf of X and Y be defined by f (x, y) = c(x + y),
x =0, 1, 2, y = 0, 1, with y ≤ x.
1. Find g(x | y) and draw a figure depicting the conditional
pmfs for y =0 and 1.
2. Find h(y | x) and draw a figure depicting the conditional
pmfs for x = 0, 1 and2.
3. Find P(0 < X <2 |Y = 0), P(X ≤ 2 |Y...
Let the random variable and have the joint pmf X Y f(x,y) =
{x(y)^2}/c
where x = 1, 2, 3 ; y = 1, 2, x+y<= 4, that is (x,y) are {(1,1),
(1,2), (2,1), (2,2), 3,1)}
(a) Find . c > 0
(b) Find . μX
(c) Find . μY
(d) Find . σ2 X
(e) Find . σ2 Y
(f) Find Cov . (X,Y )
(g) Find p , Corr . (x,y)
(h) Are and X and Y independent
The joint pmf of ? and ? is given by ??,? (?, ?) = (? + ?)/ 27
??? ? = 0, 1,2; ? = 1, 2, 3, and ??,? (?, ?) = 0 otherwise. a. Find
?(?|? = ?) for all ? = 0,1, 2. b. Find ?(3 + 0.2?|? = 2).
Let the random variable X and Y have the joint pmf f(x, y) = , c
xy 2 where x = 1, 2, 3; y = 1, 2, x + y ≤ 4 , that is, (x, y) are
{(1, 1),(1, 2),(2, 1),(2, 2),(3, 1)} .
(a) Find c > 0 .
(b) Find μ . X
(c) Find μ . Y
(d) Find σ . 2 X
(e) Find σ . 2 Y
(f) Find Cov (X, Y )...
The joint pmf of (X,Y) is depicted below.
f(x,y)
y=0
y-=1
y=2
y=3
x=0
0.02
0.05
0.06
0.12
x=1
0.03
0.12
0.15
0.13
x=2
0.02
0.10
0.15
0.05
a.) What is the marginal pmf of X.
b.) Calculate E(X).
c.) What is the conditional pmf (probability mass function) of X
given that Y = 1.
d.) Calculate E(X | Y = 1).
e.) Calculate Var(X | Y = 1).
f.) What is the cov(X,Y)? Note cov means
covariance
Define the joint pmf of (X, Y) by
f(0, 10) = f(0, 20) = 1 / 24, f(1, 10) = f(1, 30) = 1 /
24,
f(1, 20) = 6 / 24, f(2, 30) = 14 / 24
Find the value of the following. Give your answer to three
decimal places.
a) E(Y | X = 0) =
b) E(Y | X = 1) =
c) E(Y | X = 2) =
d) E(Y) =
SOLUTION REQUIRED WITH COMPLETE STEPS
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=0)
a) Find the value of ?
b) Find the marginal pmf of ? and ?
c) Find conditional pmf of ? given ? = 2