Suppose that X₁, X2, X3,...,Xn 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). n-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.
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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) = 0According 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).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.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.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) = ?
- 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 (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;Suppose X1,…,Xn,…are identically distributed with mean E(X1)=μ<∞ and Var(X1)=σ2<∞. In addition, we assume that Cov(Xk,Xk+1)=0 for k=1,2,… but Cov(Xk,Xj)=0 whenever ∣k−j∣≥2. (a) Find the limiting distribution ofXˉn=n−1i=1∑nXiasn→∞. (b) Find the limiting distribution of Zn=∑nnXin+∑n=1nXiX,eias n→∞. (c) LetY1,Y2… be i.i.d random variables with mean 0 and variance 1 . Additionally, letXk=Yk+Yk+1 for ,k≥1. Find the limiting distribution of Xˉn⋅If F(x1, x2, x3) is the value of the joint distribution function of X1, X2, and X3 at (x1, x2, x3), show that the joint marginal distribution function of X1 and X3 is given by M(x1, x3) = F(x1, q, x3) for −q < x1 < q, −q < x3 < q and that the marginal distribution function of X1 is given by G(x1) = F(x1, q, q) for −q < x1 < q With reference to Example 19, use these results to find (a) the joint marginal distribution function of X1 and X3; (b) the marginal distribution function of X1.