Question

Let {Xi} be i.i.d. random variables with P(Xi=−1) = P(Xi= 1) = 1/2. Let Sn= 1 +X1+. . .+Xn be symmetric simple random walk with initial point S0 = 1. Find the probability that Sn eventually hits the point 0.

Hint: Define the events A={Sn= 0 for some n} and for M >1, AM = {Sn hits 0 before hitting M}.

Show that AM ↗ A.

Answer #1

X1, … Xn are i.i.d. random variables, and E(Xi ) = 3β, Var(Xi )
= 3β^2 , i = 1 … n, β > 0. Two estimators of β are defined as β̂
1 = (X̅ /3) β̂ 2 = (n /3n+1 ) X̅
Show that MSE(β̂ 2) < MSE(β̂ 1) for a sample size of n =
3.

Let {Xj} ∞ j=1 be a collection of i.i.d. random variables
uniformly distributed on [0, 1]. Let N be a Poisson random variable
with mean n, and consider the random points {X1 , . . . , XN }.
b. Let 0 < a < b < 1. Let C(a,b) be the number of the
points {X1 , . . . , XN } that lies in (a, b). Find the conditional
mass function of C(a,b) given that N =...

Honest data is repeatedly released independently. Let Xi be the result of
i-th launch and Sn = X1 + X2,. . . , Xn, obtain:
a) lim→∞ P(Sn> 3n).
b) An approximate value for P (S100> 320)
Answers: a)1; b)0,96

Let X1, X2, ... be i.i.d. r.v. and N an
independent nonnegative integer valued r.v. Let
SN=X1 +...+ XN.
Assume that the m.g.f. of the Xi, denoted
MX(t), and the m.g.f. of N, denoted MN(t) are
finite in some interval (-δ, δ) around the origin.
1. Express the m.g.f. MS_N(t) of SN in terms
ofMX(t) and MN(t).
2. Give an alternate proof of Wald's identity by computing the
expectation E[SN] as M'S_N(0).
3. Express the second moment E[SN2] in terms...

Let Xi, i = 1, 2..., 48, be independent random variables that
are uniformly distributed on the interval [-0.5, 0.5].
(a) Find the probability Pr(|X1|) < 0.05
(b) Find the approximate probability P (|Xbar| ≤ 0.05).
(c) Determine an approximation of a such that P(Xbar ≤ a) =
0.15

(14pts) Let X and Y be i.i.d. geometric random variables with
parameter (probability of success) p, 0 < p < 1. (a) (6pts)
Find P(X > Y ). (b) (8pts) Find P(X + Y = n) and P(X = k∣X + Y =
n), for n = 2, 3, ..., and k = 1, 2, ..., n − 1.

2 Let X1,…, Xn be a sample of iid NegBin(4, ?) random variables
with Θ=[0, 1]. Determine the MLE ? ̂ of ?.

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 i.i.d. with N(theta, 1)
a) find the CR Rao lower-band for the variance of an
unbiased estimator of theta
b)------------------------------------of theta^2
c)-----------------------------------of P(X>0)

Let S4 = X1 + ... + X4 is the sum of 4 independent random
variables, and each Xi is exponential with
λ = 3. Find the moment generating function m(t) for S4. Also
find m′(0) and m′′(0）

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