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])

Expert Solution

orchestra answered 3 years ago

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).

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 uniform random variables on [0, 1]. If X and Y are...

Let X and Y be uniform random variables on [0, 1]. If X and Y are independent, find the probability distribution function of X + Y

Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1,...

Let X and Y be random variables. Suppose P(X = 0, Y = 0) = .1, P(X = 1, Y = 0) = .3, P(X = 2, Y = 0) = .2 P(X = 0, Y = 1) = .2, P(X = 1, Y = 1) = .2, P(X = 2, Y = 1) = 0. a. Determine E(X) and E(Y ). b. Find Cov(X, Y ) c. Find Cov(2X + 3Y, Y ).

1. Assume that X and Y are two independent discrete random variables and that X~N(0,1) and...

1. Assume that X and Y are two independent discrete random variables and that X~N(0,1) and Y~N(µ,σ2). a. Derive E(X3) and deduce that E[((Y-µ)/σ)3] = 0 b. Derive P(X > 1.65). With µ = 0.5 and σ2 = 4.0, find z such that P(((Y-µ)/σ) ≤ z) = 0.95. Does z depend on µ and/or σ? Why

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?

X and Y are discrete random variables with joint distribution given below Y 1 Y 0...

X and Y are discrete random variables with joint distribution given below Y 1 Y 0 Y 1 X 1 0 1/4 0 X 0 1/4 1/4 1/4 (a) Determine the conditional expectation E Y|X 1 . (b) Determine the conditional expectation E X|Y y for each value of y. (c) Determine the expected value of X using conditional expectation results form part (b) above. (d) Now obatin the marginal distribution of X and verify your answers to part (c).

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 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).

Question