Exercise 1. Establish the following logical equivalencies where
the domain of P(x) is non-empty and Adoes...
Exercise 1. Establish the following logical equivalencies where
the domain of P(x) is non-empty and Adoes not depend upon x:
i) ∀x(A → P(x)) ≡ A → ∀xP(x).
ii) ∀x(P(x) → A) ≡ ∃xP(x) → A.
A geometric distribution has a pdf given by P(X=x) = p(1-p)^x,
where x = 0, 1, 2, ..., and 0 < p < 1. This form of the
geometric starts at x = 0, not at x = 1. Given are the following
properties:
E(X) = (1-p)/p, and Var(X) = (1-p)/p^2
A random sample of size n is drawn; the data are X1, X2, ...,
Xn.
A. Derive the Fisher information function for the parameter
p.
B. Find the Cramér-Rao...
1)
a. Which of the following sets are the empty
set?
i. { x | x is a real number and x2 – 1 = 0 }
ii. { x | x is a real number and x2 + 1 = 0 }
iii. { x | x is a real number and x2 = -9 }
iv. { x | x is a real number and x = 2x + 1 }
b. Let A = {1, 2, 3,...
let R be a ring; X a non-empty set and (F(X, R), +, *) the ring
of the functions from X to R. Show directly the associativity of
the multiplication of F(X, R). Assume that R is unital and
commutative. show that F(X, R) is also unital and commutative.
Complete this formal proof of Ex(P(x)v~P(x)) from the empty set.
NOTE: similar to the rule above when instantiating quantifiers, if
you need a random name, always start at the beginning of the
alphabet. That is, use a first; only use b if necessary; etc.
Let
X1,X2,...,Xn
be i.i.d. (independent and identically distributed) from the
Bernoulli distribution
f(x)=p^x(1-p)^1-x,
x=0,1,p∈(0,1) where p is unknown parameter. Find the UMVUE of
p parameter and calculate MSE (Mean Square Error) of this UMVUE
estimator.
2. Let X ~ Geometric (p) where 0 < p <1
a. Show explicitly that this family is “very regular,” that is,
that R0,R1,R2,R3,R4 hold.
R 0 - different parameter values have different functions.
R 1 - parameter space does not contain its own endpoints.
R 2. - the set of points x where f (x, p) is not zero and should
not depend on p.
R 3. One derivative can be found with respect to p.
R 4. Two...
Let X ~ Geometric (p) where 0 < p <1
a) Show explicitly that this family is “very regular,” that is,
that R0,R1,R2,R3,R4 hold.
R 0 - different parameter values have different functions.
R 1 - parameter space does not contain its own endpoints.
R 2. - the set of points x where f (x, p) is not zero and should
not depend on p.
R 3. One derivative can be found with respect to p.
R 4. Two derivatives...