Question

- Let
*X**1**,**X**2**,...,**X**n*be i.i.d. (independent and identically distributed) from the uniform distribution*U(μ,μ+1)*where*μ∈R*is unknown. Find a minimal sufficient statistic for*μ*parameter.

Answer #1

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, ....

Suppose that (X1, · · · , Xn) is a random sample from uniform
distribution U(0, θ).
(a) Prove that T(X1, · · · , Xn) = X(n) is minimal sufficient
for θ. (X(n) is the largest order statistic, i.e., X(n) = max{X1, ·
· · , Xn}.)
(b) In addition, we assume θ ≥ 1. Find a minimal sufficient
statistic for θ and justify your answer.

Suppose that X1, X2, . . . , Xn are independent identically
distributed random
variables with variance σ2. Let Y1 = X2 +X3 , Y2 = X1 +X3 and
Y3 = X1 + X2. Find the following : (in terms of σ2)
(a) Var(Y1)
(b) cov(Y1 , Y2 )
(c) cov(X1 , Y1 )
(d) Var[(Y1 + Y2 + Y3)/2]

let X1 X2 ...Xn-1 Xn be independent exponentially distributed
variables with mean beta
a). find sampling distribution of the first order statistic
b). Is this an exponential distribution if yes why
c). If n=5 and beta=2 then find P(Y1<=3.6)
d). find the probability distribution of Y1=max(X1, X2, ...,
Xn)

Let X1, X2, · · · , Xn (n ≥ 30)
be i.i.d observations from N(µ1,
σ12 ) and Y1, Y2, · · ·
, Yn be i.i.d observations from N(µ2,
σ22 ). Also assume that X's and Y's are
independent. Suppose that µ1, µ2,
σ12 ,
σ22 are unknown. Find an
approximate 95% confidence interval for
(µ1µ2).

You are given that X1 and X2 are two independent and identically
distributed random variables with a Poisson distribution with mean
2. Let Y = max{X1, X2}. Find P(Y = 1).

If I know that (X1, X2, ...Xn) is an i.i.d sample of an
exponential distribution, how can I get a distribution of theta
(the parameter for the pdf, or the expectation of the distribution)
with x bar?

let X1, . . . , Xn i.i.d. Gamma(α, β), β > 0 known and α >
0 unknown. (a) Find the sufficient statistic for α (b) Use the
sufficient statistic found in (a) to find the MVUE of α n .
b) Use the sufficient statistic found in (a) to find the MVUE of
α n

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 iid
random variables from a gamma distribution with unknown α and
unknown β. Find the method of moments estimators for α and β

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