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....
X and Y are independent Exponential random variables with
mean=4, λ = 1/2.
1) Find the joint CDF of the random variables X, Y
and Find the probability that 4X > Y .
2) Find the expected value of X^3 + X*Y .
Let X and Y be independent continuous random variables, with
each one uniformly distributed in the interval from 0 to1. Compute
the probability of the following event.
XY<=1/7
Let X and Y be two independent random variables. X is a binomial
(25,0.4) and Y is a uniform (0,6). Let W=2X-Y and Z= 2X+Y.
a) Find the expected value of X, the expected value of Y, the
variance of X and the variance of Y.
b) Find the expected value of W.
c) Find the variance of W.
d) Find the covariance of Z and W.
d) Find the covariance of Z and W.
Let X and Y be independent Gaussian(0,1) random variables.
Define the random variables R and Θ, by R2=X2+Y2,Θ = tan−1(Y/X).You
can think of X and Y as the real and the imaginary part of a
signal. Similarly, R2 is its power, Θ is the phase, and R is the
magnitude of that signal.
(b) Find the probability density functions of R and Θ. Are R and
Θ independent random variables?
Let X and Y be independent positive random variables. Let Z=X/Y.
In what follows, all occurrences of x, y, z are assumed to be
positive numbers.
Suppose that X and Y are discrete, with known PMFs, pX and pY.
Then,
pZ|Y(z|y)=pX(?).
What is the argument in the place of the question mark?
Suppose that X and Y are continuous, with known PDFs, fX and fY.
Provide a formula, analogous to the one in part (a), for fZ|Y(z|y)
in terms...
If X and Y are independent exponential random
variables, each having parameter λ = 6, find the joint
density function of U = X + Y and
V = e 2X.
The required joint density function is of the form
fU,V (u, v)
=
{
g(u, v)
u > h(v), v >
a
0
otherwise
(a)
Enter the function g(u, v) into the
answer box below.
(b)
Enter the function h(v) into the answer box
below.
(c)
Enter the value...