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The random variable X is uniformly distributed in the interval [0, α] for some α > 0.

Parameter α is fixed but unknown. In order to estimate α, a random sample X1, X2, . . . , Xn of independent and identically distributed random variables with the same distribution as X is collected, and the maximum value Y = max{X1, X2, ..., Xn} is considered as an estimator of α.

(a) Derive the cumulative distribution function of Y .

(b) Find the mean and variance of Y , and explain why Y is a good estimator for α when sample size n is large.

Answer #1

The random variable X is uniformly distributed in the interval
[0, α] for some α > 0. Parameter α is fixed but unknown. In
order to estimate α, a random sample X1, X2, . . . , Xn of
independent and identically distributed random variables with the
same distribution as X is collected, and the maximum value Y =
max{X1, X2, ..., Xn} is considered as an estimator of α.
(a) Derive the cumulative distribution function of Y .
(b)...

Consider n independent variables, {X1, X2, . . . , Xn} uniformly
distributed over the unit interval, (0, 1). Introduce two new
random variables, M = max (X1, X2, . . . , Xn) and N = min (X1, X2,
. . . , Xn).
(A) Find the joint distribution of a pair (M, N).
(B) Derive the CDF and density for M.
(C) Derive the CDF and density for N.
(D) Find moments of first and second order for...

Let Y1,Y2.....,Yn be independent ,uniformly distributed random
variables on the interval[0,θ].，Y(n)=max(Y1,Y2,....,Yn)，which is
considered as an estimator of θ. Explain why Y is a good estimator
for θ when sample size is large.

6. Let X1, X2, ..., Xn be a random sample of a random variable X
from a distribution with density
f (x) ( 1)x 0 ≤ x ≤ 1
where θ > -1. Obtain,
a) Method of Moments Estimator (MME) of parameter θ.
b) Maximum Likelihood Estimator (MLE) of parameter θ.
c) A random sample of size 5 yields data x1 = 0.92, x2 = 0.7, x3 =
0.65, x4 = 0.4 and x5 = 0.75. Compute ML Estimate...

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).

Suppose that X1,..., Xn form a random sample from the
uniform distribution on the interval [0,θ], where the value of the
parameter θ is unknown (θ>0).
(1)What is the maximum likelihood estimator of θ?
(2)Is this estimator unbiased? (Indeed, show that it underestimates
the parameter.)

A uniform random variable on (0,1), X, has density function f(x)
= 1, 0 < x < 1. Let Y = X1 + X2 where X1 and X2 are
independent and identically distributed uniform random variables on
(0,1).
1) By considering the cumulant generating function of Y ,
determine the first three cumulants of Y .

Suppose that X is uniformly distributed on the interval [0,5], Y
is uniformly distributed on the interval [0,5], and Z is uniformly
distributed on the interval [0,5] and that they are
independent.
a)find the expected value of the max(X,Y,Z)
b)what is the expected value of the max of n independent random
variables that are uniformly distributed on [0,5]?
c)find pr[min(X,Y,Z)<3]

The random variable X is distributed with pdf fX(x,
θ) = (2/θ^2)*x*exp(-(x/θ)2), where x>0 and
θ>0. Please note the term within the exponential is
-(x/θ)^2 and the first term includes a θ^2.
a) Find the distribution of Y = (X1 + ... +
Xn)/n where X1, ..., Xn is an
i.i.d. sample from fX(x, θ). If you can’t find Y, can
you find an approximation of Y when n is large?
b) Find the best estimator, i.e. MVUE, of θ?

Suppose that X is a random variable uniformly distributed over
the interval (0, 2), and Y is a random variable uniformly
distributed over the interval (0, 3). Find the probability density
function for X + Y .

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