Question

Is the following a probability mass function (probability distribution function)? Show why or why not.

P(X=x) = 3 / (4 *(3-X)! *X!) for x = 0, 1, 2, 3

and zero elsewhere

**What do they mean by, "and zero elsewhere", other than
that I understand the question. Thanks!**

Answer #1

TOPIC:Conditions for being a valid pmf.

Complete the following two problems
(a) Show that P(X = x) =(1/2)^ x+1 for x = 0,1,2,... is a valid
probability mass function for the discrete random variable X.
(b) Given the
following cumulative distribution function, nd the probability mass
function.
x
P(X ≤ x)
0
0.05
1
0.32
2
0.64
3
0.95
4
1

Suppose that X follows geometric distribution with the
probability of success p, where 0 < p < 1, and probability of
failure 1 − p = q. The pmf is given by f(x) = P(X = x) = q
x−1p, x = 1, 2, 3, 4, . . . . Show that X is a pmf.
Compute the mean and variance of X without using moment generating
function technique. Show all steps.
To compute the variance of X, first compute...

6. You have been provided the following probability
distribution:
Pets
P(x)
x*P(x)
(x-µ)²P(x)
0
0.330
1
0.270
2
0.300
3
0.050
4
0.025
5
0.020
6 or more
0.005
Let x = the total number of pets in a
household
a. Complete table
b. What is the probability of a household having no
pets?
c. What is the probability of a household having exactly 3
pets?
d. What is the probability of a household having
between 1 and 3
pets?...

(1) Consider X that follows the Bernoulli distribution with
success probability 1/4, that is, P(X = 1) = 1/4 and P(X = 0) =
3/4. Find the probability mass function of Y , when Y = X4 . Find
the second moment of Y . (2) If X ∼ binomial(10, 1/2), then use the
binomial probability table (Table A.1 in the textbook) to find out
the following probabilities: P(X = 5), P(2.9 ≤ X ≤ 4.9) (3) A deck
of...

Consider a discrete random variable X with probability mass
function P(X = x) = p(x) = C/3^x, x = 2, 3, 4, . . . a. Find the
value of C. b. Find the moment generating function MX(t). c. Use
your answer from a. to find the mean E[X]. d. If Y = 3X + 5, find
the moment generating function MY (t).

1. Find k so that f(x) is a probability density function. k=
___________
f(x)= { 7k/x^5 0 1 < x < infinity elsewhere
2. The probability density function of X is f(x).
F(1.5)=___________
f(x) = {(1/2)x^3 - (3/8)x^2 0 0 < x < 2
elsewhere
3. F(x) is the distribution function of X. Find the probability
density function of X. Give your answer as a piecewise
function.
F(x) = {3x^2 - 2x^3 0 0<x<1 elsewhere

Let X 1 and X 2 have the joint probability distribution function
??(??1, ??2) = � ??−(??1+??2) ??1 > 0, ??2 > 0 0 elsewhere
Find ??(??1 + ??2) and ??(??1 + ??2).

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).]

For each of the random quantities X,Y, and Z, defined below
(a) Plot the probability mass function PMS (in the discrete
case) , or the probability density function PDF (in the continuous
case)
(b) Calculate and plot the cumulative distribution function
CDF
(c) Calculate the mean and variance, and the moment function
m(n), and plot the latter.
The random quantities are as follows:
X is a discrete r.q. taking values k=0,1,2,3,... with probabilities
p(1-p)^k, where p is a parameter with...

Suppose that a random variable X has the distribution (pdf) f(x)
=kx(1 -x^2) for
0 < x < 1 and zero elsewhere.
a. Find k.
b. Find P(X >0. 8)
c. Find the mean of X.
d. Find the standard deviation of X.
2. Assume that test scores for all students on a statistics test
are normally
distributed with mean 82 and standard deviation 7.
a. Find the probability that a single student scores greater than
80.
b. Find the...

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