Question

Let Θ be a Bernoulli random variable that indicates which one of
two hypotheses is true, and let **P**(Θ=1)=*p*.
Under the hypothesis Θ=0, the random variable *X* has a
normal distribution with mean 0, and variance 1. Under the
alternative hypothesis Θ=1, *X* has a normal distribution
with mean 2 and variance 1.

Suppose for this part of the problem that *p*=2/3. The MAP
rule can choose in favor of the hypothesis Θ=1 if and only if
*x*≥*c*1. Find the value of *c*1.

For this part, assume again that *p*=2/3. Find the
conditional probability of error for the MAP decision rule, given
that the hypothesis Θ=0 is true. **P**(error|Θ=0)=

Find the overall (unconditional) probability of error associated
with the MAP rule for *p*=1/2.

Answer #1

a) Observe that if is the rejection region of , then

Therefore

b) The probability of error given that is

c) Now the map rule for p = 1/2 would be (from the previous calculation)

Therefore the unconditional probability of error is

Let X1, . . . , Xn be a random sample from a Bernoulli(θ)
distribution, θ ∈ [0, 1]. Find the MLE of the odds ratio, defined
by θ/(1 − θ) and derive its asymptotic distribution.

Let X_1,…, X_n be a random sample from the Bernoulli
distribution, say P[X=1]=θ=1-P[X=0].
and
Cramer Rao Lower Bound of θ(1-θ)
=((1-2θ)^2 θ(1-θ))/n
Find the UMVUE of θ(1-θ) if such exists.
can you proof [part (b) ] using (Leehmann Scheffe
Theorem step by step solution) to proof
[∑X1-nXbar^2 ]/(n-1) is the umvue , I have the key
solution below
x is complete and sufficient.
S^2=∑ [X1-Xbar ]^2/(n-1) is unbiased estimator of θ(1-θ) since
the sample variance is an unbiased estimator of the...

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 X1, X2, ·······, Xn be a random sample from the Bernoulli
distribution. Under the condition 1/2≤Θ≤1, find a
maximum-likelihood estimator of Θ.

Let X be the binomial random variable obtained by adding n=4
Bernoulli Trials, each with probability of success p=0.25. Define
Y=|X-E(x)|. Find the median of Y.
A.0
B.1
C.2
D.3
E.Does not exist

Let θ > 1 and let X1, X2, ..., Xn be a random sample from the
distribution with probability density function f(x; θ) = 1/xlnθ , 1
< x < θ.
c) Let Zn = nlnY1. Find the limiting distribution of Zn.
d) Let Wn = nln( θ/Yn ). Find the limiting distribution of
Wn.

Let Y ⇠ Gamma(alpha,beta) and conditioned on Y = y, X
⇠ Poisson(y).
Find the unconditional distribution of X in the case that alpha = r
is an integer and beta=1-p/p
for p in (0, 1).
Find the conditional
distribution of Y|X = x. (Use Bayes’ rule)

Let X1,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ is
InvGamma(10,2). (a) Find the posterior distribution for θ. (b) If
n= 10 and ̄x= 18.2, find the Bayes estimate under
squared error loss. (c) The variance of the data distribution is φ=
3θ2. Find the Bayes estimator (under squared error loss)
for φ.Let X1,...,Xn∼iid Gamma(3,1/θ) and we assume the prior for θ
is InvGamma(10,2). (a) Find the posterior distribution for θ. (b)
If n= 10 and ̄x= 18.2, find...

2. Let the probability density function (pdf) of random variable
X be given by:
f(x) = C (2x -
x²),
for
0< x < 2,
f(x) = 0,
otherwise
Find the value of
C.
(5points)
Find cumulative probability function
F(x)
(5points)
Find P (0 < X < 1), P (1< X < 2), P (2 < X
<3)
(3points)
Find the mean, : , and variance,
F².
(6points)

Problem 1. Let x be a random variable which approximately
follows a normal distribution with mean µ = 1000 and σ = 200. Use
the z-table, calculator, or computer software to find the
following: Part A. Find P(x > 1500). Part B. Find P(x < 900).
Part C. Find P(900 < x < 1500).

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