Question

1. Suppose a random variable X has a pmf

p(x) = 3^(x-1)/4^x , x = 1,2,...

(a) Find the moment generating function of X.

(b) Give a realistic example of an experiment that this random
variable can be defined from its sample space.

(c) Find the mean and variance of X.

Answer #1

Realistic example:

A company tests the manufactured bulbs until the first defective bulb is found. So the number of bulbs tested will be a geometric random variable taking values 1,2,3...

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).]

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...

Suppose that X is a discrete random variable with ?(? = 1) = ?
and ?(? = 2) = 1 − ?. Three independent observations of X are made:
(?1, ?2, ?3) = (1,2,2).
a. Estimate ? through the sample mean (this is an example of the
“method of moment” for estimating a parameter).
b. Find the likelihood function and MLE for ?.

Let N be a positive integer random variable with PMF of the form
pN(n)=12⋅n⋅2−n,n=1,2,…. Once we see the numerical value of N , we
then draw a random variable K whose (conditional) PMF is uniform on
the set {1,2,…,2n} . 1. Find joint PMF pN,K(n,k) For n=1,2,… and
k=1,2,…,2n 2. Find the marginal PMF pK(k) as a function of k . For
simplicity, provide the answer only for the case when k is an even
number. For k=2,4,6,… 3. Let...

Let X be a normal random variance with media 1 and variance 4.
Consider a new variance A random variable T defined below:
T = -1 if X < -2
T = 0 if - 2 ≤ X ≤ 0
T = 1 if x>0
Find the moment generating function of T and, from it, calculate E (T) and Var (T).

The random variables, X and Y , have the joint pmf
f(x,y)=c(x+2y), x=1,2 y=1,2 and zero otherwise.
1. Find the constant, c, such that f(x,y) is a valid pmf.
2. Find the marginal distributions for X and Y .
3. Find the marginal means for both random variables.
4. Find the marginal variances for both random variables.
5. Find the correlation of X and Y .
6. Are the two variables independent? Justify.

Let f(x) = (1/2)^x, x = 1,2,3,... be the PMF of the random
variable X. Find the MGF, mean, and variance of X.

Suppose that the moment generating function of a random variable
X is of the form MX (t) = (0.4e^t + 0.6)8 . What is the moment
generating function, MZ(t), of the random variable Z = 2X + 1?
(Hint: think of 2X as the sum two independent random variables).
Find E[X]. Find E[Z ]. Compute E[X] another way - try to recognize
the origin of MX (t) (it is from a well-known distribution)

Suppose that ? is a random variable with the following pmf: ?(?)
= ?/ ? ; ? = 1,2,3,4,5
a) Find the expected value and variance of ?.
b) Find the expected value of √?, ?[√?].

The moment generating function for the random variable X is
MX(t) = (e^t/ (1−t )) if |t| < 1. Find the variance of X.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 48 seconds ago

asked 49 seconds ago

asked 2 minutes ago

asked 3 minutes ago

asked 3 minutes ago

asked 3 minutes ago

asked 3 minutes ago

asked 4 minutes ago

asked 5 minutes ago

asked 7 minutes ago

asked 7 minutes ago

asked 11 minutes ago