Question

Let X have a uniform distribution on (0, 1) and let y = -ln ( x )

a. Construct the CDF of Y graphically

b. Find the CDF of Y using CDF method

c. Find the PDF of Y using PDF method

Answer #1

a) Graph:

Let U1 and U2 be independent Uniform(0, 1) random variables and
let Y = U1U2.
(a) Write down the joint pdf of U1 and U2.
(b) Find the cdf of Y by obtaining an expression for FY (y) =
P(Y ≤ y) = P(U1U2 ≤ y) for all y.
(c) Find the pdf of Y by taking the derivative of FY (y) with
respect to y
(d) Let X = U2 and find the joint pdf of the rv pair...

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 ?

Let X and Y have the pdf f(x, y) = 1, 0 < x < 1, 0
< y < 1, zero elsewhere.
Find the cdf and pdf of the product Z =
X+Y.

Let X ∼ UNIF(0, 1). Find the pdf of Y = −5 ln(X) using the
transformation technique. Note that Y is an exponential random
variable. What is its parameter? Show your work.

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...

Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and
0<=y<=1.
a) Find k.
b) Find the joint cumulative density function of (X,Y)
c) Find the marginal pdf of X and Y.
d) Find Pr[Y<X2] and Pr[X+Y>0.5]

X is said to have a uniform distribution between (0, c), denoted
as X ~ U(0, c), if its probability density function f(x) has the
following form
f(x) = (1/c , if 0<x<c, 0 otherwise)
(a) (2pts) Write down the pdf for X ~ U(0, 2).
(b) (3pts) Find the cumulative distribution function (cdf) F(x)
of X ~ U(0, 2).
(c) Find the mean, second moment, variance, and standard
deviation for X ~ U(0, 2).
(d) Let Y be the...

X is said to have a uniform distribution between (0, c), denoted
as X ⇠ U(0, c), if its probability density function f(x) has the
following form f(x) = (1 c , if x 2 (0, c), 0 , otherwise .
(a) (2pts) Write down the pdf for X ⇠ U(0, 2).
(b) (3pts) Find the cumulative distribution function (cdf) F(x)
of X ⇠ U(0, 2). (Caution: Please specify the function values for
all 1
(c) Find the mean, second...

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 (X; Y ) have a uniform distribution on the unit circle in
the plane.
(a) Show that X and Y are not independent.
(b) Find P(X2 + Y 2 < 1=4) .

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