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

thank you

please upvote

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 )

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.

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 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)

Let ?? ~ ??? (?,θ ) independent random variable. for ? = 1,2,…
?. Find an estimator for ? by the maximum likelihood
method.(MLE)

Suppose X is a discrete random variable that takes on integer
values between 1 and 10, with variance Var(X) = 6. Suppose that you
define a new random variable Y by observing the output of X and
adding 3 to that number. What is the variance of Y? Suppose then
you define a new random variable Z by observing the output of X and
multiplying that by -4. What is the variance of Z?

Let X be a Poisson random variable with parameter λ and Y an
independent Bernoulli random variable with parameter p. Find the
probability mass function of X + Y .

Let X be a discrete random variable with the range RX = {1, 2,
3, 4}. Let PX(1) = 0.25, PX(2) = 0.125, PX(3) = 0.125.
a) Compute PX(4).
b) Find the CDF of X.
c) Compute the probability that X is greater than 1 but less
than or equal to 3.

Suppose X_1, X_2, … X_n is a random sample from a population
with density f(x) = (2/theta)*x*e^(-x^2/theta) for x greater or
equal to zero.
Find the Maximum Likelihood Estimator Theta-Hat_1 for
theta.
Find the Method of Moment Estimator Theta-Hat_2 for theta.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 6 minutes ago

asked 6 minutes ago

asked 8 minutes ago

asked 9 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 20 minutes ago

asked 20 minutes ago

asked 24 minutes ago

asked 24 minutes ago