Question

Let *p*p be a polynomial, and let
*θ*∈(−1,1)θ∈(−1,1). Show that the series
∑∞*k*=1*p*(*k*)*θ**k*∑k=1∞p(k)θk
converges absolutely.

Answer #1

Determine if the series converges conditionally, converges
absolutely, or diverges.
/sum(n=1 to infinity) ((-1)^n(2n^2))/(n^2+4)
/sum(n=1 to infinity) sin(4n)/4^n

let x be a discrete random variable with positive integer
outputs.
show that P(x=k) = P( x> k-1)- P( X>k) for any positive
integer k.
assume that for all k>=1 we have P(x>k)=q^k. use (a) to
show that x is a geometric random variable.

suppose sigma n=1 to infinity of square root ((a_n)^2 +
(b_n)^2)) converges. Show that both sigma a_n and sigma b_n
converge absolutely.

Let 0 < θ < 1 and let (xn) be a sequence where
|xn+1 − xn| ≤ θn for n
= 1, 2, . . ..
a) Show that for any 1 ≤ n < m one has |xm −
xn| ≤ (θn/ 1-θ )*(1 − θ m−n ).
Conclude that (xn) is Cauchy
b)If lim xn = x* , prove the following error in
approximation (the "error in approximation" is the same as error
estimation in Taylor Theorem) in t:...

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 the next two series, (1) find the interval of convergence
and (2) study convergence at the end points of the interval if any.
Also, (3) indicate for what values of x the series converges
absolutely, conditionally, or not at all. You must indicate the
test you use and show the interval of convergence both analytically
and graphically and summarize your results on the picture.
∑∞ n=1 ((−1)^n−1)/ (n^1/4)) *x^n

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 Geom(p). For positive integers n, k define
P(X = n + k | X > n) = P(X = n + k) / P(X > n) :
Show that P(X = n + k | X > n) = P(X = k) and then briefly
argue, in words, why this is true for geometric random
variables.

10. Let P(k) be the following statement: ”Let a1, a2, . . . , ak
be integers and p be a prime. If p|(a1 · a2 · a3 · · · ak), then
p|ai for some i with 1 ≤ i ≤ k.” Prove that P(k) holds for all
positive integers k

Let
P0<tn-1<k=0.42
, and k>0 and n≥2 and n is an
integer. Find the following:
P-k<tn-1<k
P-k<tn-1<0
Ptn-1<k

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