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 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 be a continuous random variable with pdf: f(x) = ax^2 −
2ax, 0 ≤ x ≤ 2
(a) What should a be in order for this to be a legitimate
p.d.f?
(b) What is the distribution function (c.d.f.) for X?
(c) What is Pr(0 ≤ X < 1)? Pr(X > 0.5)? Pr(X > 3)?
(d) What is the 90th percentile value of this distribution?
(Note: If you do this problem correctly, you will end up with a
cubic...
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 be two continuous random variables
with the joint probability density function of for 0 < x
< 2, 0 < y < 2, x + y < 1,where c is a
constant. (In all the following answers, you do NOT need to find
what the value of c is; just treat it as a number.)
(a) Write out the marginal distribution of Y.
(b) P(Y < 1/3) = ?
(c) P(X < 1.5, Y < 0.5)=...
Let X and Y be independent Exponential random variables with
common mean 1.
Their joint pdf is f(x,y) = exp (-x-y) for x > 0 and y > 0
, f(x, y ) = 0 otherwise. (See "Independence" on page 349)
Let U = min(X, Y) and V = max (X, Y).
The joint pdf of U and V is f(u, v) = 2 exp (-u-v) for 0 < u
< v < infinity, f(u, v ) = 0 otherwise....
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 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)
2. Let X be a continuous random variable with PDF ?fx(x)= cx(1 −
x), 0 < x < 1,
0 elsewhere.
(a) Find the value of c such that fX(x) is indeed a PDF.
(b) Find P(−0.5 < X < 0.3).
(c) Find the median of X.