i need the answer quickly Q₂ let X₁, X₂-Xn denote random samplefrom adistribution with P. d. f.8-10x8.f(x)0) =OS X (1elsewhereOwhere Oso, using the factori Zation theorem,Show That Tn = #x;, is a sufficient statistic✔for 0.(211560 that th CPa

Answered: Image /qna-images/answer/793515d7-7a38-43da-92e8-cd73d8e21d6b.jpg

Answered: where Oso, using the factors Zation…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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The random variable X has a Bernoulli distribution with parameter p. A random sampleX1, X2, . . . , Xn of size n is taken of X. Show that the sample proportionX1 + X2 + · · · + Xnnis a minimum variance unbiased estimator of p.Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population withparameter p, show thatY =n∑j=1
Let X1, ..., Xn be a random sample of size n from a gamma population given by the density f(x; α, β) = ( x α−1e − x β βαΓ(α) if x > 0, α > 0, β > 0 0 if x ≤ 0 with Γ(α) = R ∞ 0 x α−1 e −xdx the well-known gamma distribution. 1. If α > 0 is known for this random sample compute the maximum likelihood estimator βbML estimator of the unknown parameter β > 0. 2. Compute the moment generating function E(e sX) for every s < 1 β . 3. Compute the expectation E(X). 4. Show that the maximum likelihood estimator βbML is a unbiased estimator of β.(Hint: you may use the result of part 2)
7. Assume that X₁,..., Xn is a random sample from a Bernoulli (p) and let Yn n 1/1 X₁. For p # 1/2, Determine the asymptotic distribution of √n[Yn(1 – Yn) - p(1 - p)]. 7. Assume that \( X_1, ..., X_n \) is a random sample from a Bernoulli(p) and let \( {Y}_n = \1/n\sum_{i=1}^{n} X_i \). For \( p =/ 1/2), Determine the asymptotic distribution of \( \sqrt{n}({Y}_n(1 - {Y}_n) - p(1 - p)) \).
Consider a random sample X1, … , Xn from the pdff (x; u) = .5(1 + (THETA)x) -1 <= x <= 1where -1 <= theta <= 1 (this distribution arises in particlephysics). Show that theta = 3X is an unbiased estimator oftheta. [Hint: First determine mu = E(X) = E(X).]
A stochastic process (SP) X(t) is given byX(t) = Asin(ωt + Φ)where A and Φ are independent random variables and Φ is uniformly distributed between 0 and 2π.a) Calculate mean E[X(t)]. b) Calculate the auto-correlation RX (t1,t2).c) Is X(t) wide sense stationary (WSS)? Justify your answer.Now consider that X(t) is a Gaussian SP with mean μX (t) = 0.5 and auto-correlation RX (t1,t2) =10e−14 |t1−t2|. Let Z = X(5) and W = X(9) be the two random variables.d) Calculate var(Z), var(W), and var(Z + W). e) Calculate cov(ZW).
Let Yn denote the nth order statistic of a random sample of size n froma distribution of the continuous type. Find the smallest value of n for which theinequality P(ξ0.9 < Yn) ≥ 0.75 is true.
A researcher is interested in testing whether annual house hold income in Philadelphia is normal. So she took a sample of 50 house holds and found that skewness (s)= 2.3190 and Kurtosis (k) = 6.7322. Use the Jarque- Bera Test to test , at alpha 0.05, whether income follows normal distribution. -Yes, population is normal because Chi-Square test is higher than critical value. -Yes, population is normal because Chi-Square test is less than critical value. -No, population is not normal because Chi-Square test is higher than critical value. -No, population is not normal because Chi-Square test is less than critical value.
T2 #4 Apr 5 Of the international passengers arriving at an airport, 1.5% are selected for luggage inspection. Let X be the number of passengers from a random sample of 400 who are sent for luggage inspection. Use Poisson approximation to X to find the probability that between 6 and 12 passengers inclusive, are sent for luggage inspection.
A random variable follows a distribution of the form f(x)=k(x+2)e^-x over x>orequalto 0 . Determine the probability that two independent samples are drawn from the population and they both have that 1 < x < 2. State your answer in exact form (with a bunch of e’s), showing all work.
Let X1, X2, ..., Xn be a sequence of independent and identically distributedrandom variables having the Exponential(λ) distribution, λ > 0,fXi(x) = λe−λx , x > 00 , otherwise(a) Show that the moment generating function mX(s) := E(e^sX) = λ/λ−s for s < λ;(b) Using (a) find the expected value E(Xi) and the variance Var(Xi).(c) Define the random variable Y = X1 + X2 +· · ·+ Xn. Find E(Y ), Var(Y ) and the moment generating function of Y .(d) Consider a random variable X having Gamma(α, λ) distribution,fX(x) = (λαxα-1/Γ(α)) e−λx , x > 00 , otherwiseShow that the moment generating function of the random variable X is mX(s) =λα 1/(λ−s)α for s < λ, where Γ(α) isΓ(α) = (integral from 0 to inifity ) xα−1e−xdx.(e) What is the probability distribution of Y given in (c)? Explain youranswer.
Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?
LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.

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where Oso, using the factors Zation theorem, Tn = #x;, is a sufficient statisti Show That for 0. (21