The joint probability distribution of variables X and
Y is shown in the table below.
...............................................................................X.......................................................................
Y
1
2
3
1
0.30
0.18
0.12
2
0.15
0.09
0.06
3
0.05
0.03
0.02
Calculate E(XY)
Determine the marginal probability distributions of X
and Y.
Calculate E(X) and E(Y)
Calculate V(X) and V(Y)
Are X and Y independent? Explain.
Find P(Y = 2| X = 1)
Calculate COV(X,Y). Did you expect
this answer? Why?
Find the probability distribution...
The joint probability distribution of random variables, X and Y,
is shown in the following table: X 2 4 6 Y 1 0.10 0.20 0.08 2 0.06
0.12 0.16 3 0.15 0.04 0.09
(a) Calculate P ( X=4 | Y=1)
(b) Calculate V (Y | X=2) .
(c) Calculate V (3Y-X ) .
Let X and Y have joint discrete distribution p(x, y) = 3 20 (.5
x ) (.7 y ), x = 0, 1, 2, . . . , and y = 0, 1, 2, . . .. Find the
marginal probability function P(X = x). [hint: for a geometric
series X∞ n=0 arn with −1 < r < 1, r 6= 0, then X∞ n=0 arn =
a 1 − r ]
4. The joint density function of (X, Y ) is
f(x,y)=2(x+y), 0≤y≤x≤1
. Find the correlation coefficient ρX,Y
.
5. The height of female students in KU follows a normal
distribution with mean 165.3 cm and s.d. 7.3cm. The height of male
students in KU follows a normal distribution with mean 175.2 cm and
s.d. 9.2cm. What is the probability that a random female student is
taller than a male student in KU?
Let X and Y have the joint pdf f(x, y) = 8xy, 0 ≤ x ≤ y ≤ 1. (i)
Find the conditional means of X given Y, and Y given X. (ii) Find
the conditional variance of X given Y. (iii) Find the correlation
coefficient between X and Y.
Let X and Y have the following joint distribution:
X/Y
0
1
2
0
5/50
8/50
1/50
2
10/50
1/50
5/50
4
10/50
10/50
0
Further, suppose σx = √(1664/625), σy = √(3111/2500)
a) Find Cov(X,Y)
b) Find p(X,Y)
c) Find Cov(1-X, 10+Y)
d) p(1-X, 10+Y), Hint: use c and find Var[1-X], Var[10+Y]
Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and
0<=y<=1.
a) Find k.
b) Find the joint cumulative density function of (X,Y)
c) Find the marginal pdf of X and Y.
d) Find Pr[Y<X2] and Pr[X+Y>0.5]
Suppose the joint probability distribution of X and Y is given
by the following table.
Y=>3 6 9 X
1 0.2 0.2 0
2 0.2 0 0.2
3 0 0.1 0.1
The table entries represent the probabilities. Hence the
outcome [X=1,Y=6] has probability
0.2.
a) Compute E(X), E(X2), E(Y), and E(XY). (For all answers show
your work.) b) Compute E[Y | X = 1], E[Y | X = 2], and E[Y | X =
3].
c) In this case, E[Y...
Let X and Y have the following joint density function
f(x,y)=k(1-y) , 0≤x≤y≤1.
Find the value of k that makes this a probability density
function.
Compute the probability that P(X≤3/4, Y≥1/2).
Find E(X).
Find E(X|Y=y).