Question

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 ?.

Answer #1

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The range of a discrete random variable X is {−1, 0, 1}. Let MX
(t) be the moment generating function of X, and let MX(1) = MX(2) =
0.5. Find the third moment of X, E(X^3).

The range of a discrete random variable X is {−1, 0, 1}. Let
MX(t) be the moment generating function of X, and let MX(1) = MX(2)
= 0.5. Find the third moment of X, E(X^3 )

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

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.

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).

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.

(i) If a discrete random variable X has a moment generating
function
MX(t) = (1/2+(e^-t+e^t)/4)^2, all t
Find the probability mass function of X. (ii) Let X and Y be two
independent continuous random variables with moment generating
functions
MX(t)=1/sqrt(1-t) and MY(t)=1/(1-t)^3/2, t<1
Calculate E(X+Y)^2

Suppose the random variable X follows the Poisson P(m) PDF, and
that you have a random sample X1, X2,...,Xn from it. (a)What is the
Cramer-Rao Lower Bound on the variance of any unbiased estimator of
the parameter m? (b) What is the maximum likelihood estimator
ofm?(c) Does the variance of the MLE achieve the CRLB for all
n?

Suppose X is a discrete random variable with probability mass
function given by
p (1) = P (X = 1) = 0.2
p (2) = P (X = 2) = 0.1
p (3) = P (X = 3) = 0.4
p (4) = P (X = 4) = 0.3
a. Find E(X^2) .
b. Find Var (X).
c. Find E (cos (piX)).
d. Find E ((-1)^X)
e. Find Var ((-1)^X)

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...

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