Question

In: Statistics and Probability

Let X and Y be independent random variables with mean 𝜇X and 𝜇𝑌, and variances 𝜎𝑋 2 and 𝜎𝑌 2 respectively

Let X and Y be independent random variables with mean 𝜇X and 𝜇𝑌, and variances 𝜎𝑋 2 and 𝜎𝑌 2 respectively. Show that

𝑉𝑎𝑟[𝑋 ∙ 𝑌] = 𝜎𝑋 2 ∙ 𝜎𝑌 2 + 𝜇𝑌 2 ∙ 𝜎𝑋 2 + 𝜇𝑋 2 ∙ 𝜎𝑌 2

Expert Solution

GIVEN:

Let X and Y be independent random variables with mean and , and variances and respectively.

TO PROVE:

PROOF:

Given the mean of X

Variance of X

The formula for variance is given by,

Given the mean of Y

Variance of Y

The formula for variance is given by,

Thus we have ; ; ; ; and .

First, we compute the mean of XY,

{Since X and Y are independent.}

Now the formula for variance of XY is given by,

{Since X and Y are independent and .}

Using and , we obtain the result,

Thus .

orchestra answered 2 years ago

Let X and Y be independent Poisson random variables with parameters 1 and 2, respectively, compute...

Let X and Y be independent Poisson random variables with parameters 1 and 2, respectively, compute P(X=1 and Y=2) P(X+Y>=2) Find Poisson approximations to the probabilities of the following events in 500 independent trails with probabilities 0.02 of success on each trial. 1 success 2 or fewer success.

Let X, Y be independent exponential random variables with mean one. Show that X/(X + Y...

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 two independent random variables such that X + Y has the...

Let X and Y be two independent random variables such that X + Y has the same density as X. What is Y?

Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2,...

Let X, Y be independent random variables with X ∼ Uniform([1, 5]) and Y ∼ Uniform([2, 4]). a) FindP(X<Y). b) FindP(X<Y|Y>3) c) FindP(√Y<X<Y).

Let X and Y be two independent random variables. X is a binomial (25,0.4) and Y...

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, and Z independent random variables with variance 4 and mean 1. Find the...

Let X, Y, and Z independent random variables with variance 4 and mean 1. Find the correlation coefficient between (X-2YX+1) and (4X+Y)

Let X and Y be independent Exponential random variables with common mean 1. Their joint pdf...

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 X and Y be two independent random variables. Assume that X is Negative- Binomial(2, θ)...

Let X and Y be two independent random variables. Assume that X is Negative- Binomial(2, θ) and Y is Negative-Binomial(3, θ) distributed. Let Z be another random variable, Z = X + Y . (a) Find the following probabilities: P(Z = 0), P(Z = 1) and P(Z = 2); (b) Can you guess what is the distribution of Z?

Let X and Y be independent Gaussian(0,1) random variables. Define the random variables R and Θ,...

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...

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...