Question

We are given a random number generator that generates a uniform random variable X over the interval [0, 1]. Suppose we flip a fair coin and if H occurs, we report X and if T occurs, we report 2X + 1. Let Y be the reported random variable.

(a) Derive the cdf and pdf of Y

(b) Which one is more likely to occur: Y ∈ [0,1] or Y ∈ [1,2]? Explain your answer.

Answer #1

We are given a random number generator that generates a uniform
random variable X over the interval [0,1]. Suppose we ﬂip a fair
coin and if H occurs, we report X and if T occurs, we report 2X +
1. Let Y be the reported random variable.
(a) Derive the cdf and pdf of Y [15 points].
(b) Which one is more likely to occur: Y ∈ [0,1] or Y ∈ [1,2]?
Explain your answer [10 points].

The CDF of a discrete random variable
X is given by
F(x) = [x(x+1)(2x+1)]/ [n(n+1)(2n+1)],
x =1,2,….n.
Derive the probability mass function.
Show that it is a
valid probability mass function.

a) Suppose that X is a uniform continuous random variable where
0 < x < 5. Find the pdf f(x) and use it to find P(2 < x
< 3.5).
b) Suppose that Y has an exponential distribution with mean 20.
Find the pdf f(y) and use it to compute P(18 < Y < 23).
c) Let X be a beta random variable a = 2 and b = 3. Find P(0.25
< X < 0.50)

Let X be a continuous uniform (-2,5) random variable. Let W =
|X| Your goal is to find the pdf of W.
a)Begin by finding the sample space of W
b)Translate the following into a probability statement about X:
Fw(w) = P[W <= w] = ....
c) Consider different values of W the sample of W. Do you need
to break up the sample space into cases?
d)Find the cdf of W
e)Find the pdf of W

Let
the random variable X have pdf
f(x) = x^2/18; -3 < x < 3 and zero otherwise.
a) Find the pdf of Y= X^2
b) Find the CDF of Y= X^2
c) Find P(Y<1.9)

Let
X be a continuous random variable rv distributed via the pdf f(x)
=4e^(-4x) on the interval [0, infinity].
a) compute the cdf of X
b) compute E(X)
c) compute E(-2X)
d) compute E(X^2)

Suppose a random variable X has cumulative distribution function
(cdf) F and probability
density function (pdf) f. Consider the random variable Y =
X?b
a for a > 0 and real b.
(a) Let G(x) = P(Y x) denote the cdf of Y . What is the
relationship between the functions
G and F? Explain your answer clearly.
(b) Let g(x) denote the pdf of Y . How are the two functions f
and g related?
Note: Here, Y is...

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...

Question 3 Suppose the random variable X has the uniform
distribution, fX(x) = 1, 0 < x < 1. Suppose the random
variable Y is related to X via Y = (-ln(1 - X))^1/3.
(a) Demonstrate that the pdf of Y is fY (y) = 3y^2 e^-y^3,
y>0. (Hint: Work out FY (y))
(b) Determine E[Y ]. (Hint: Use Wolfram Alpha to undertake the
integration.)

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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