Question

Let
X and Y be random variable follow uniform U[0, 1]. Let Z = X to the
power of Y. What is the distribution of Z?

Answer #1

Let random variable X ∼ U(0, 1). Let Y = a + bX, where a and b
are constants.
(a) Find the distribution of Y .
(b) Find the mean and variance of Y .
(c) Find a and b so that Y ∼ U(−1, 1).
(d) Explain how to find a function (transformation), r(), so
that W = r(X) has an exponential distribution with pdf f(w) = e^
−w, w > 0.

Let X and Y be i.i.d and follow a uniform distribution in [0,1].
Find the joint distribution of(U,V) where U=X+Y and V =X/Y.

Let X, Y ∼ U[0, 1], be independent and let Z = max{X, Y }. (a)
(10 points) Calculate Pr[Z ≤ a]. (b) (10 points) Calculate the
density function of Z. (c) (5 points) Calculate V ar(Z).

Let X and Y be independent random variables, with X following
uniform distribution in the interval (0, 1) and Y has an Exp (1)
distribution.
a) Determine the joint distribution of Z = X + Y and Y.
b) Determine the marginal distribution of Z.
c) Can we say that Z and Y are independent? Good

Let X and Y be independent random variables each having the
uniform distribution on [0, 1].
(1)Find the conditional densities of X and Y given that X > Y
.
(2)Find E(X|X>Y) and E(Y|X>Y) .

Let U = Z+ ∪ { 0 }.
Let R1 = { ( x, y ) | x ≠ y }
Let R2 = { ( x, y ) | x = y }
Let R3 = { ( x, y ) | x ≥ y }
Let R4 = { ( x, y ) | x ≤ y }
Let R5 = { ( x, y ) | x > y }
Let R6 = { ( x, y...

Suppose that X and Y are independent Uniform(0,1) random
variables. And let U = X + Y and V = Y .
(a) Find the joint PDF of U and V
(b) Find the marginal PDF of U.

Let X follow Poisson distribution with λ = a and Y follow
Poisson distribution with λ = b. X and Y are independent. Define a
new random variable as Z=X+Y. Find P(Z=k).

Let ? be a random variable with a PDF
?(?)= 1/(x+1) for ? ∈ (0, ? − 1). Answer the following
questions
(a) Find the CDF
(b) Show that a random variable ? = ln(? + 1) has uniform ?(0,1)
distribution. Hint: calculate the CDF of ?

Assume X and. Y are. 2. independent variables that follow the
standard uniform distribution i.e. U(0,1)
Let Z = X + Y
Find the PDF of Z, fZ(z) by first obtaining the CDF
FZ(z) using the following steps:
(a) Draw an x-y axis plot, and sketch on this plot the lines
z=0.5, z=1, and z=1.5 (remembering z=x+y)
(b) Use this plot to obtain the function which describes the
area below the lines for z = x + y in terms...

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