Question

Let two independent random vectors x and z have Gaussian distributions: p(x) = N(x|µx,Σx), and p(z) = N(z|µz,Σz). Now consider y = x + z. Use the results for Gaussian linear system to ﬁnd the distribution p(y) for y. Hint. Consider p(x) and p(y|x). Please prove for it rather than directly giving the result.

Answer #1

Uncorrelated and Gaussian does not imply independent unless
jointly Gaussian. Let X ∼N(0,1) and Y = WX, where p(W = −1) = p(W =
1) = 0 .5. It is clear that X and Y are not independent, since Y is
a function of X. a. Show Y ∼N(0,1). b. Show cov[X,Y ]=0. Thus X and
Y are uncorrelated but dependent, even though they are Gaussian.
Hint: use the deﬁnition of covariance cov[X,Y]=E [XY] −E [X] E [Y ]
and...

Independent random variables X and Y follow binomial
distributions with parameters(n1,θ) and (n2,θ). Let Z =X+Y. What
will be the distribution of Z?
Hint: Use moment generating function.

You have two random variables X and Y
X -> μX = 5 , σX = 3
Y -> μY = 7 , σY = 4
Now, we define two new random variables
Z = X - Y
W = X + Y
Answer the below questions:
μZ =
[ Select ]
["3",
"1", "-2"]
σZ =
...

Let X and Y be random variables with the following distributions
X N~ (20,2) and Y N~ (30,1). The covariance of X and Y is σ XY =
0.25 . Let Z= + 0.75X x 0.25Y . Find the mean and the variance of
Z.

Problems
1. Two independent random variables X and Y
have the probability distributions as follows:
X 1 2 5
P (X) 0.2 0.5 0.3
Y 2 4
P (Y) 0.7 0.3
a) Let T = X + Y. Find all possible values of T.
Compute μ and . T σ T
b) Let U = X - Y. Find all possible values of U.
Compute μ U and σ U .
c) Show that μ T
= μ X +...

Let X and Y be independent random variables following Poisson
distributions, each with parameter λ = 1. Show that the
distribution of Z = X + Y is Poisson with parameter λ = 2. using
convolution formula

Assume that X~N(0, 1), Y~N(0, 1) and X and Y are independent
variables.
Let Z = X+Y, and joint density of Y and Z is expressed as f(y,
z) = g(z|y)*h(y)
g(z|y) is conditional distribution of Z given y, and h(y) is
density of Y
how can i get f(y, z)?

Let X and Y be two independent random variables with
μX =E(X)=2,σX =SD(X)=1,μY =2,σY =SD(Y)=3.
Find the mean and variance of
(i) 3X
(ii) 6Y
(iii) X − Y

Let X ~ N(1,3) and Y~ N(5,7) be two independent random
variables. Find...
Var(X + Y + 32)
Var(X -Y)
Var(2X - 4Y)

7.
Let X and Y be two independent and identically distributed
random variables with expected value 1 and variance 2.56.
(i) Find a non-trivial upper bound for
P(| X + Y -2 | >= 1)
(ii) Now suppose that X and Y are independent and identically
distributed N(1;2.56) random variables. What is P(|X+Y=2| >= 1)
exactly? Briefly, state your reasoning.
(iii) Why is the upper bound you obtained in Part (i) so
different from the exact probability you obtained in...

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