Let X and Y have the joint pdf f(x, y) = 8xy, 0 ≤ x ≤...
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 joint PDF
f(x) = c(e^-(x/λ + y/μ)) 0 < x < infinity and 0 < y
< infinity
with parameters λ > 0 and μ > 0
a) Find c such that this is a PDF.
b) Show that X and Y are Independent
c) What is P(1 < X < 2, 0 < Y < 5) ? Leave in
exponential form
d) Find the marginal distribution of Y, f(y)
e) Find 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 the continuous random variables X and Y have the joint
pdf:
f(x,y)=6x , 0<x<y<1
i) find the marginal pdf of X and Y respectively,
ii) the conditional pdf of Y given x, that is
fY|X(y|x),
iii) E(Y|x) and Corr(X,Y).
Let X and Y be continuous random variables with joint pdf
f(x, y) = kxy^2 0 < x, 0 < y, x
+ y < 2
and 0 otherwise
1) Find P[X ≥ 1|Y ≤ 1.5]
2) Find P[X ≥ 0.5|Y ≤ 1]
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
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?
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)
Let X and Y be two jointly continuous random variables with
joint PDF
f(x,y) = Mxy^2
0<x<y<1
a) Find M = ?
b) Find the marginal probability densities.
c) P( y> 1/2 | x = .25) = ?
d) Corr (x,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).