Let X ∼Exp(1), Y ∼Exp(2) be independent random variables. (a) What is the range of Z...

Let X ∼Exp(1), Y ∼Exp(2) be independent random variables.

(a) What is the range of Z := X + Y ?

(b) Find the pdf of Z.

(d) Let U = e Y . What is the range of U?

(e) Find the pdf of U|X.

Solutions

Expert Solution

orchestra answered 1 year ago

Next > < Previous

Related Solutions

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...

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, 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).

Suppose X and Y are independent random variables with Exp(θ = 2) distribution. Note that, we...

Suppose X and Y are independent random variables with Exp(θ = 2) distribution. Note that, we say X ∼ Exp(θ) if its pdf is f(x) = 1/θ e^(−x/θ) , for x > 0 and θ > 0. (a) What is the joint probability density function (pdf) of (X, Y )? (b) Use the change of variable technique (transformation technique) to evaluate the joint pdf fW,Z (w, z) of (W, Z), where W = X −Y and Z = Y ....

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, 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 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 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 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.

1. Let X and Y be independent random variables with μX= 5, σX= 4, μY= 2,...

1. Let X and Y be independent random variables with μX= 5, σX= 4, μY= 2, and σY= 3. Find the mean and variance of X + Y. Find the mean and variance of X – Y. 2. Porcelain figurines are sold for $10 if flawless, and for $3 if there are minor cosmetic flaws. Of the figurines made by a certain company, 75% are flawless and 25% have minor cosmetic flaws. In a sample of 100 figurines that are...

Subjects