Question

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.

Answer #1

TOPIC:Finding the pmf from the cdf.

Suppose that the probability mass function for a discrete random
variable X is given by p(x) = c x, x = 1, 2, ... , 9. Find the
value of the cdf (cumulative distribution function) F(x) for 7 ≤ x
< 8.

Let the probability density function of the random variable X be
f(x) = { e ^2x if x ≤ 0 ;1 /x ^2 if x ≥ 2 ; 0 otherwise}
Find the cumulative distribution function (cdf) of X.

A random variable X has the cumulative distribution function
(cdf) given by F(x) = (1 + e−x ) −1 , −∞ <
x < ∞.
(i) Find the probability density function (pdf) of X.
(ii) Roughly, take 10 points in the range of x (5 points below 0
and 5 points more than 0) and plot the pdf on these 10 points. Does
it look like the pdf is symmetric around 0?
(iii) Also, find the expected value of X.

1. Let X be a discrete random variable with the probability mass
function P(x) = kx2 for x = 2, 3, 4, 6.
(a) Find the appropriate value of k.
(b) Find P(3), F(3), P(4.2), and F(4.2).
(c) Sketch the graphs of the pmf P(x) and of the cdf F(x).
(d) Find the mean µ and the variance σ 2 of X. [Note: For a
random variable, by definition its mean is the same as its
expectation, µ = E(X).]

Q6/
Let X be a discrete random variable defined by the
following probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Give P(4≤ X < 8)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q7/
Let X be a discrete random variable defined by the following
probability function
x
2
3
7
9
f(x)
0.15
0.25
0.35
0.25
Let F(x) be the CDF of X. Give F(7.5)
ــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ
Q8/
Let X be a discrete random variable defined by the following
probability function :
x
2
6...

1. Given a discrete random variable, X , where the
discrete probability distribution for X is given on right,
calculate E(X)
X
P(X)
0
0.1
1
0.1
2
0.1
3
0.4
4
0.1
5
0.2
2. Given a discrete random variable, X , where the
discrete probability distribution for X is given on right,
calculate the variance of X
X
P(X)
0
0.1
1
0.1
2
0.1
3
0.4
4
0.1
5
0.2
3. Given a discrete random variable, X...

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.

Let (X, Y ) have joint cdf F, and let G be the cdf of the random
variable X + Y . Show that F(x, x) ≤ G(2x) for all x ∈ R

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].

Find Std(X) in each case, when X is a discrete random
variable with the following given densities.
(i) f(x)=1/5, x=6,7,8,9,10.
(ii) f(x)=1/100f(x), x=1,2,3,⋯,100.
(iv) f(x)=0.3(0.7)^x, x=0,1.
(v) f(x)=1x/(x+1), x=1,2,

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