Let the joint p.d.f f(x,y) = 1 for 0 <= x <= 2, 0 <= y
<= 1, 2*y <= x. (And 0 otherwise)
Let the random variable W = X + Y.
Without knowing the p.d.f of W, what interval of w values holds
at least 60% of the probability?
Assume that X and Y has a continuous joint p.d.f. as
(28x^2)*(y^3) in 0<y<x<1
interval. Otherwise the joint p.d.f. is equal to 0.
Prove that the mentioned f(x,y) is a joint probability density
function.
Calculate E(X)
Calculate E(Y)
Calculate
E(X2)
Calculate Var(X)
Calculate E(XY)
Calculate P(X< 0.1)
Calculate P(X> 0.1)
Calculate P(X>2)
Calculate P(-2<X<0.1)
Suppose X and Y have joint probability density function f(x,y) =
6(x-y) when 0<y<x<1 and f(x,y) = 0 otherwise.
(a) Indicate with a sketch the sample space in the x-y plane
(b) Find the marginal density of X, fX(x)
(c) Show that fX(x) is properly normalized, i.e., that it
integrates to 1 on the sample space of X
(d) Find the marginal density of Y, fY(y)
(e) Show that fY(y) is properly normalized, i.e., that it
integrates to 1 on...
Question 6. Suppose the joint pdf of X and Y is
f(x,y) =
ax^2y for 0 < x < y 0 < y < 1
0 otherwise
Find a.
Find the correlation between X and Y.
Are X and Y independent? Explain.
Find the conditional variance Var(X||Y = 1)
Given:
f(x,y) = 5 - 3x - y for 0 < x,y < 1 and x + y < 1, 0
otherwise
1) find the covariance of x and y
2) find the marginal probability density function for x
c) find the probability that x >= 0.6 given that y <=
0.2
The joint PDF of X and Y is given by f(x, y) = C, (0<
x<y<1).
a) Determine the value of C
b) Determine the marginal distribution of X and compute E(X) and
Var(X)
c) Determine the marginal distribution of Y and compute E(Y) and
Var(Y)
d) Compute the correlation coefficient between X and Y
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) =
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,...