a)Suppose that X₁, X2, X3.....X₁ is a random sample of n from population X which has a chi-square distribution with one degrees of freedom.(i) Find lim Mg(t).71-00(ii) Use the Central Limit Theorem to compute P(0.5 < X < 1.5), for n = 30.(iii) Use the Chebychev inequality to find the value of k for P(0.5 < X < 1.5), if n = 30.

Given Xi~chi square distribution with one degree of freedom, and they are random sample…

Answered: a) Suppose that X₁, X₂, X3....X₁ is a…

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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Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0
According to the Central Limit Theorem, the distribution of sample means is NOT normal if _________________. A) The population from which the samples are selected is not normally distributed and the sample size is small (n < 30). B) The population from which the samples are selected is not normally distributed and the sample size is large (n >= 30). C) The population from which the samples are selected is normally distributed and the sample size is small (n < 30). D) The population from which the samples are selected is normally distributed and the sample size is large (n >= 30).
1.9.5. Let a random variable $X$ of the continuous type have a pdf $f(x)$ whose graph is symmetric with respect to $x=c .$ If the mean value of $X$ exists, show that $E(X)=c$Hint: Show that $E(X-c)$ equals zero by writing $E(X-c)$ as the sum of two integrals: one from $-\infty$ to $c$ and the other from $c$ to $\infty .$ In the first, let $y=c-x$ and, in the second, $z=x-c .$ Finally, use the symmetry condition $f(c-y)=f(c+y)$ in the first.
Let Xi ∼ Bernoulli(1/2), and let Yn = X1 + X2 + · · · + Xn. Use the central limit theorem to findP(Y100 < 60).
Consider a birth and death process, X = {X(t)} : t ≥ 0 with the 4 states, S = {0, 1, 2, 3} and instant rates per hour are specified implicitly as follows: 1 /λk = 10 minutes for (0 ≤ k ≤ 2) and 1/ µk = 30 minutes for (1 ≤ k ≤ 3) Assume that λk = 0 for k ≥ 3 and µk = 0 for k > 3. a) Derive the limiting distribution, lim t→ ∞ P [X(t) = k] for all k b) Evaluate expectations in minutes for departure times, E [S1 |X(0) = 0] and E [S3 |X(0) = 3]
Let X1, . . . , Xn ∼ iid Unif(0, θ). (a) Derive an asymptotic distribution for the MOM estimator ˜θ = 2X¯ of the form.(b) From this choose an approximate pivot and interval to get an interval that asymptotically has100(1 − α)% coverage.
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.
Let X be an exponential random variable with standard deviation σ. FindP(|X − E(X)| > kσ ) for k = 2, 3, 4, and compare the results to the boundsfrom Chebyshev’s inequality.
The interval [0,5] is partitioned into nn equal subintervals, and a number xi is arbitrarily chosen in the ith subinterval for each i. Then: lim (n→∞) n ∑ i=1 (2xi−2/n) = ?
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