1 Let f(x, y) = 4xy, 0 < x < 1, 0 < y < 1, zero
elsewhere, be the joint probability density function(pdf) of X and
Y . Find P(0 < X < 1/2 , 1/4 < Y < 1) , P(X = Y ), and
P(X < Y ). Notice that P(X = Y ) would be the volume under the
surface f(x, y) = 4xy and above the line segment 0 < x = y <
1...
Let f(x, y) be a function such that f(0, 0) = 1, f(0, 1) = 2,
f(1, 0) = 3, f(1, 1) = 5, f(2, 0) = 5, f(2, 1) = 10. Determine the
Lagrange interpolation F(x, y) that interpolates the above data.
Use Lagrangian bi-variate interpolation to solve this and also show
the working steps.
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?
Let f (x, y) = c, 0 ≤ y ≤ 4, y ≤ x ≤ y + 1, be the
joint pdf of X and Y.
(a) (3 pts) Find c and sketch the region for which f (x, y) >
0.
(b) (3 pts) Find fX(x), the marginal pdf of X.
(c) (3 pts) Find fY(y), the marginal pdf of Y.
(d) (3 pts) Find P(X ≤ 3 − Y).
(e) (4 pts) E(X) and Var(X).
(f) (4 pts) E(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]
Let X and Y have the joint PDF
f(x) = { 1/2 0 < x + y < 2, x > 0, y > 0 ;
{ 0 elsewhere
a) sketch the support of X and Y
b) Are X and Y independent? Explain.
c) Find P(x<1 and y<1.5)
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 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).