Question

We can work with factorial moments to calculate the moments of discrete distributions.

a.) Calculate the factorial moment of E(X(X-1)) for the binomial and poisson distributions. Then use the results to find the variances of the binomial and poisson distributions.

Answer #1

: The Root Mean
Square of a discrete-time signal is given by
We can easily
calculate the RMS of signals in MATLAB using a combination of the
sum, sqrt, and ^ commands. N represents the number of samples of
the signal. Please note that the command .^ applied to a vector
squares each element of the vector.
Write a script to
calculate the RMS of the discrete-time signal x, defined as
follows: n = a vector of number between -23...

Problem 3 (m.g.f. of Binomial random variables). We proved that
the m.g.f. ψX(t) “generates” the moments of the random variable X
by differentiation, and computation at t = 0 (rather than by
integration or summation, which is typically harder). For example,
ψ 0 X(0) = E(X), ψ00 X(0) = E(X 2 ), . . . , ψ(n) X (0) = E(X n ),
where the superscript (n) indicates the n th derivative. (a) Assume
that X ∼ Binomial(n, p), i.e....

Problem 1: Relations among Useful Discrete Probability
Distributions. A Bernoulli experiment consists of
only one trial with two outcomes (success/failure) with probability
of success p. The Bernoulli distribution
is
P (X = k) =
pkq1-k,
k=0,1
The sum of n independent Bernoulli trials forms a binomial
experiment with parameters n and p. The binomial probability
distribution provides a simple, easy-to-compute approximation with
reasonable accuracy to hypergeometric distribution with parameters
N, M and n when n/N is less than or equal...

You must decide which of two discrete distributions a random
variable X has. We will call the distributions
p0 and p1. Here are the
probabilities they assign to the values x of
X:
x
0
1
2
3
4
5
6
p0
0.1
0.1
0.2
0.3
0.1
0.1
0.1
p1
0.1
0.3
0.2
0.1
0.1
0.1
0.1
You have a single observation on X and wish to test
H0: p0 is correct
Ha: p1 is correct
One possible decision procedure...

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 )

QUESTION 1
The expected value of a discrete random variable is:
A)The mean
B)The standard deviation
C) The probability of success
D) The variance
QUESTION 2
Expected value is:
A)A measure of dispersion
B) A measure of central location
C)A measure of distance from the mean
D)None of the above
QUESTION 3
In combinations,
A)n represents the number of objects, x represents multiply
B)n represents nothing, x represents the number of elements
C)n represents the total number of objects, x...

(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

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).

Method of Moments Concept Question II
1 point possible (graded)
Let (E,{Pθ}θ∈Θ) denote a statistical model associated to a
statistical experiment X1,…,Xn∼iidPθ∗ where θ∗∈Θ is the true
parameter. Assume that Θ⊂Rd for some d≥1. Let mk(θ):=E[Xk] where
X∼Pθ. mk(θ) is referred to as the k-th moment of
Pθ . Also define the moments map:
ψ:Θ
→Rd
θ
↦(m1(θ),m2(θ),…,md(θ)).
What conditions on ψ do we have to assume so that the method of
moments produces a consistent and asymptotically normal estimator?...

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