Let (A,B) have joint PDF f(a,b)=(ca^2b^2 when 0 < a,b,a+b
< 1 and 0 otherwise for...
Let (A,B) have joint PDF f(a,b)=(ca^2b^2 when 0 < a,b,a+b
< 1 and 0 otherwise for some constant c > 0. 1. Find a
formula for E[A | B = b]. 2. Find Cov(A,B).
Let Y1 and Y2 have joint pdf f(y1, y2) = (6(1−y2), if 0≤y1≤y2≤1
0, otherwise. a) Are Y1 and Y2 independent? Why? b) Find Cov(Y1,
Y2). c) Find V(Y1−Y2). d) Find Var(Y1|Y2=y2).
Let ?1 and ?2 have the joint pdf
f (?1, ?2)= 6?2 0<?2<?1<1
=0 else where
A. Find conditional mean and conditional variance ?1given?2
.
B. Theorem of total mean and total variance?1given?2
.(urgently needed)
Let X1 and X2 have the joint pdf
f(x1,x2) = 2 0<x1<x2<1; 0. elsewhere
(a) Find the conditional densities (pdf) of X1|X2 = x2 and X2|X1
= x1.
(b) Find the conditional expectation and variance of X1|X2 = x2 and
X2|X1 = x1.
(c) Compare the probabilities P(0 < X1 < 1/2|X2 = 3/4) and
P(0 < X1 < 1/2).
(d) Suppose that Y = E(X2|X1). Verify that E(Y ) = E(X2), and that
var(Y ) ≤ var(X2).
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 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]
Let X and Y have the joint probability density function (pdf):
f(x, y) = 3/2 x2(1 − y), − 1 < x < 1, − 1 < y < 1
Find P(0 < Y < X).
Find the respective marginal pdfs of X and Y. Are X and Y independent?
Find the conditional pdf of X give Y = y, and E(X|Y = 0.5).
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 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 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)