Find the marginal distributions of random variables X,Y, and Z given they are uniformly distributed on...

Find the marginal distributions of random variables X,Y, and Z given they are uniformly distributed on a unit sphere.

Expert Solution

orchestra answered 1 year ago

Let X, Y and Z be independent random variables, each uniformly distributed on the interval (0,1)....

Let X, Y and Z be independent random variables, each uniformly distributed on the interval (0,1). (a) Find the cumulative distribution function of X/Y. (b) Find the cumulative distribution function of XY. (c) Find the mean and variance of XY/Z.

Let X and Y be independent and uniformly distributed random variables on [0, 1]. Find the...

Let X and Y be independent and uniformly distributed random variables on [0, 1]. Find the cumulative distribution and probability density function of Z = X + Y.

Let X and Y be uniformly distributed independent random variables on [0, 1]. a) Compute the...

Let X and Y be uniformly distributed independent random variables on [0, 1]. a) Compute the expected value E(XY ). b) What is the probability density function fZ(z) of Z = XY ? Hint: First compute the cumulative distribution function FZ(z) = P(Z ≤ z) using a double integral, and then differentiate in z. c) Use your answer to b) to compute E(Z). Compare it with your answer to a).

Let X and Y be independent continuous random variables, with each one uniformly distributed in the...

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 Y and Z be independent continuous random variables, both uniformly distributed between 0 and 1....

Let Y and Z be independent continuous random variables, both uniformly distributed between 0 and 1. 1. Find the CDF of |Y − Z|. 2. Find the PDF of |Y − Z|.

9.8 Let X and Y be independent random variables with probability distributions given by P(X =...

9.8 Let X and Y be independent random variables with probability distributions given by P(X = 0) = P(X = 1) = 1/2 and P(Y = 0) = P(Y = 2) = 1/2 . a. Compute the distribution of Z = X + Y . b. Let Y˜ and Z˜ be independent random variables, where Y˜ has the same distribution as Y , and Z˜ the same distribution as Z. Compute the distribution of X˜ = Z˜ − Y

Suppose that z = xy, where x and y are independent and normally distributed random variables

Suppose that z = xy, where x and y are independent and normally distributed random variables. The mean and variance of x are µx = 10 and σ2x = 2. The mean and variance of y are µy = 15 and σ2y = 3. Find the mean and variance of z by simulation. Does µz = µxµy? Does σ2z = σ2x σ2y? Do this for 100, 1000, and 5000 trials.

The joint density function for random variables X, Y, and Z is f(x, y, z)= Cxyz if...

The joint density function for random variables X, Y, and Z is f(x, y, z)= Cxyz if 0 ≤ x ≤ 1, 0 ≤ y ≤ 2, 0 ≤ z ≤ 2, and f(x, y, z) = 0 otherwise. (a) Find the value of the constant C. (b) Find P(X ≤ 1, Y ≤ 1, Z ≤ 1). (c) Find P(X + Y + Z ≤ 1).

Assume that X, Y, and Z are independent random variables and that each of the random...

Assume that X, Y, and Z are independent random variables and that each of the random variables have a mean of 1. Further, assume σX = 1, σY = 2, and σZ = 3. Find the mean and standard deviation of the following random variables: a. U = X + Y + Z b. R = (X + Y + Z)/3 c. T = 2·X + 5·Y d. What is the correlation between X and Y? e. What is the...

Let X and Z be two independently distributed standard normal random variables and let Y=X2+Z. iShow...

Let X and Z be two independently distributed standard normal random variables and let Y=X2+Z. iShow thatE(Y|X) =X2 iiShow thatμY= 1 iiiShow thatE(XY) = 0ivShow that cov(X,Y) = 0 and thus ρX,Y= 0

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