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, Y be independent exponential random variables with mean
one. Show that X/(X + Y ) is uniformly distributed on [0, 1].
(Please solve it with clear explanations so that I can learn it.
I will give thumbs up.)
Let X and Y be continuous random variables with E[X] = E[Y] = 4
and var(X) = var(Y) = 10. A new random variable is defined as: W =
X+2Y+2. a. Find E[W] and var[W] if X and Y are independent. b. Find
E[W] and var[W] if E[XY] = 20. c. If we find that E[XY] = E[X]E[Y],
what do we know about the relationship between the random variables
X and Y?
Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,
P(X = 1, Y = 0) = .3, P(X = 2, Y = 0) = .2 P(X = 0, Y = 1) = .2,
P(X = 1, Y = 1) = .2, P(X = 2, Y = 1) = 0.
a. Determine E(X) and E(Y ).
b. Find Cov(X, Y )
c. Find Cov(2X + 3Y, Y ).
Let X and Y be jointly normal random variables with parameters
E(X) = E(Y ) = 0, Var(X) = Var(Y ) = 1, and Cor(X, Y ) = ρ. For
some nonnegative constants a ≥ 0 and b ≥ 0 with a2 +
b2 > 0 define W1 = aX + bY and W2 = bX + aY .
(a)Show that Cor(W1, W2) ≥ ρ
(b)Assume that ρ = 0.1. Are W1 and W2 independent or not?
Why?
(c)Assume now...
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...
Let X and Y be random variables with means µX and µY . The
covariance of X and Y is given by, Cov(X, Y ) = E[(X − µX)(Y − µY
)]
a) Prove the following three equalities: Cov(X, Y ) = E[(X −
µX)Y ] = E[X(Y − µY )] = E(XY ) − µXµY
b) Suppose that E(Y |X) = E(Y ). Show that Cov(X, Y ) = 0 (hint:
use the law of interated expectations to show...