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].

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 ?

The random-number generator on calculators randomly generates a
number between 0 and 1. The random variable X, the number
generated, follows a uniform probability distribution.
(a) Identify the graph of the uniform density function.
(b) What is the probability of generating a number between
0.740.74
and
0.910.91?
(c) What is the probability of generating a number greater
than
0.930.93?

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 uniform random variable on (0,1), X, has density function f(x)
= 1, 0 < x < 1. Let Y = X1 + X2 where X1 and X2 are
independent and identically distributed uniform random variables on
(0,1).
1) By considering the cumulant generating function of Y ,
determine the first three cumulants of 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.

Suppose X is a continuous uniform random variable between −1 and
1, i.e., X ∼ U(−1, 1). Find the CDF and the PDF of P = −ln|X|.

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

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
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)

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