Question

Consider the family of distributions with pmf pX(x) = p if x = −1, 2p if x = 0, 1 − 3p if x = 1 . Here p is an unknown parameter, and 0 ≤ p ≤ 1/3. Let X1, X2, . . . , Xn be iid with common pmf a member of this family. Consider the statistics A = the number of i with Xi = −1, B = the number of i with Xi = 0, C = the number of i with Xi = 1.

Write down the joint pmf of X1, X2, · · · , Xn.

Answer #1

Solution

Since X1, X2, . . . , Xn are iid, their joint probability = product of their individual probabilities.

There are A Xi’s whose value is – 1 for which probability is p.
Hence, joint probability = p^{A}.

Similarly, for B Xi’s whose value is 0 for which probability is
2p, joint probability = (2p)^{B}.

and for C Xi’s whose value is 0 for which probability is (1 –
3p), joint probability = (1 - 3p)^{C}.

Thus, joint probability pmf of X1, X2, . . . , Xn is:

**(2 ^{B}(p^{A+B})(1 -
3p)^{C},** A + B + C = n and 0 ≤ p ≤ 1/3.

**DONE**

A. (i) Consider the random variable X with pmf: pX (−1) = pX (1)
= 1/8, pX (0) = 3/4.
Show that the Chebyshev inequality P (|X − μ| ≥ 2σ) ≤ 1/4 is
actually an equation for
this random variable.
(ii) Find the pmf of a different random variable Y that also takes
the values {−1, 0, 1}
for which the Chebyshev inequality P (|X − μ| ≥ 3σ) ≤ 1/9 is
actually an equation.

(a) It is given that a random variable X such that P(X =−1) =P(X
= 1) = 1/4, P(X = 0) = 1/2. Find the mgf of X, mX(t)
(b) Let X1 and X2 be two iid random varibles such that P(Xi
=1)=P(Xi =−1)=1/2, i=1,2. Use the mgfs to prove that X and Y
=(X1+X2)/2 have the same distribution

Let X1, X2, . . . , Xn be iid following exponential distribution
with parameter λ whose pdf is f(x|λ) = λ^(−1) exp(− x/λ), x > 0,
λ > 0.
(a) With X(1) = min{X1, . . . , Xn}, find an unbiased estimator
of λ, denoted it by λ(hat).
(b) Use Lehmann-Shceffee to show that ∑ Xi/n is the UMVUE of
λ.
(c) By the definition of completeness of ∑ Xi or other tool(s),
show that E(λ(hat) | ∑ Xi)...

Let X ∼ Geo(?) with Θ = [0,1].
a) Show that pdf of the random variable X is in the
one-parameter
regular exponential family of distributions.
b) If X1, ... , Xn is a sample of iid Geo(?) random variables
with
Θ = (0, 1), determine a complete minimal sufficient statistic
for ?.

For X1, ..., Xn iid Unif(0, 1):
a) Show the conditional pdf X(i)|X(j) ∼ X(j)Beta(i, j − i)
b Let n=5, find the joint pdf between X(2) and X(4).

Let X1, . . . , Xn be i.i.d from pmf f(x|λ) where f(x) =
(e^(−λ)*(λ^x))/x!, λ > 0, x = 0, 1, 2
a) Find MoM (Method of Moments) estimator for λ
b) Show that MoM estimator you found in (a) is minimal
sufficient for λ
c) Now we split the sample into two parts, X1, . . . , Xm and
Xm+1, . . . , Xn. Show that ( Sum of Xi from 1 to m, Sum...

Let X1, X2, . . ., Xn be independent, but not identically
distributed, samples. All these Xi ’s are assumed to be normally
distributed with
Xi ∼ N(θci , σ^2 ), i = 1, 2, . . ., n,
where θ is an unknown parameter, σ^2 is known, and ci ’s are
some known constants (not all ci ’s are zero). We wish to estimate
θ.
(a) Write down the likelihood function, i.e., the joint density
function of (X1, ....

Let X = ( X1, X2, X3, ,,,, Xn ) is iid,
f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b
< 1
then,
Show the density of the statistic T = X(n) is given by
FX(n) (x) = n/ab * (x/a)^{n/(b-1}} for 0 <= x <=
a ; otherwise zero.
# using the following
P (X(n) < x ) = P (X1 < x, X2 < x, ,,,,,,,,, Xn < x
),
Then assume...

Let X =( X1,
X2, X3 ) have the joint pdf
f(x1, x2,
x3)=60x1x22, where
x1 + x2 + x3=1 and
xi >0 for i = 1,2,3. find the
distribution of X1 ? Find
E(X1).

Let X1. ..., Xn, be a random sample from Exponential(β) with pdf
f(x) = 1/β(e^(-x/β)) I(0, ∞)(x), B > 0 where β is an unknown
parameter. Find the UMVUE of β^2.

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