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.

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orchestra answered 2 years ago

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|.

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 U and V be independent continuous random variables uniformly distributed from 0 to 1. Let...

Let U and V be independent continuous random variables uniformly distributed from 0 to 1. Let X = max(U, V). What is Cov(X, U)?

1) Let U1, U2, ... be independent random variables, each uniformly distributed over the interval (0,...

1) Let U1, U2, ... be independent random variables, each uniformly distributed over the interval (0, 1]. These random variables represent successive bigs on an asset that you are trying to sell, and that you must sell by time = t, when the asset becomes worthless. As a strategy, you adopt a secret number \Theta and you will accept the first offer that's greater than \Theta . The offers arrive according to a Poisson process with rate \lambda = 1....

Let X and Y be independent discrete random variables with the following PDFs: x 0 1...

Let X and Y be independent discrete random variables with the following PDFs: x 0 1 2 f(x)=P[X=x] 0.5 0.3 0.2 y 0 1 2 g(y)= P[Y=y] 0.65 0.25 0.1 (a) Show work to find the PDF h(w) = P[W=w] = (f*g)(w) (the convolution) of W = X + Y (b) Show work to find E[X], E[Y] and E[W] (note that E[W] = E[X]+E[Y])

1. Let X and Y be independent U[0, 1] random variables, so that the point (X,...

1. Let X and Y be independent U[0, 1] random variables, so that the point (X, Y) is uniformly distributed in the unit square. Let T = X + Y. (a) Find P( 2Y < X ). (b). Find the CDF F(t) of T (for all real numbers t). HINT: For any number t, F(t) = P ( X <= t) is just the area of a part of the unit square. (c). Find the density f(t). REMARK: For a...

Let X and Y be two independent and identically distributed random variables with expected value 1...

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). 5 MARKS (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. 2 MARKS (iii) Why is the upper bound you obtained in Part (i) so different...

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.

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