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7. Let Xn Xi++X2, where the Xi's are iid standard normal random variables (a) Show that Sn is a c...
- Let {Xn} denote a sequence of iid random variables such that P(Xi = 1) = P(X1 = -1) = 1/2. Let Sn = X1 + X2 + ... + xn. (a) Find ES, and var(Sn); (b) Show that Sn is a martingale.
Let {Xn} be a sequence of iid random variables 1. (20 points) Let {Xn} be a sequence of iid random variables with common pdf f(x) = - =e-x2/2,x ER. Then find the limit in probability of the sequence of random variables {Y} where Yo: 31x11. i=1
The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2) , M(t) = R. t 2 Suppose Xi, X2, are iid random variables with this distribution. Let Sn -Xi+ (a) Show that Var(X) =3/2, i = 1,2. (b) Give the MGF of Sn/v3n/2. (c) Evaluate the limit of the MGF in (b) for n → 0. The moment generating function (MGF) for a certain probability distribution is given by 2 (2 + 2)...
Let Xi, . . . , Xn be IID random variables with mean μ, standard deviation σ and finite fourth moment. Prove by induction the identity
7. Let X1 and X2 be two iid exp(A) random variables. Set Yi Xi - X2 and Y2 X + X2. Determine the joint pdf of Y and Y2, identify the marginal distributions of Yi and Y2, and decide whether or not Yi and Y2 are independent [10)
Let Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n
Suppose that Xi, X2Xn are iid random variables with pdf eter a. (b) Find the Fisher Information of X1,X2, ,Xn and use it to estimate a 95% confidence interval on the MLE of a. (c) Explain how the central limit theorem relates to (b).
Let X1, X2, ..., Xn be iid random variables with a "Rayleigh” density having the following pdf: 22 -12 10 f(x) = e x > 0 > 0 0 пе a) (3 points) Find a sufficient estimator for 0 using the Factorization Theorem. b) (3 points) Find a method of moments estimator for 0. Small help: E(X1) = V c) (7 points) What is the MLE of 02 + 0 – 10 ? d) (7 points) For a fact, 21–1...
1. The random variables Xi, X2,.. are independent and identically distributed (iid), each with pdf f given in Assignment 4, Question 1. Let Sn- Xi+.+X Using the Central Limit Theorem and the graph of the standard normal distribution in Figure 1, approximate the probability P(S100 >600). Express your answer in the format x.x-10-x. Verify your answer by simulating 10,000 outcomes of Si00 and counting how many of them are > 600. Show the code 1.00 0.95 0.90 0.85 1.2 1.4...
1. Let Xi, X2,... be independent random variables each with the standard normal distribution, and for each n 2 0 let Sn-1 Xi. Use importance sampling to obtain good estimates for each of the following probabilities: (a) Pfmaxn<100 Sn> 10; and (b) Pímaxns100 Sn > 30) HINTS: The basic identity of importance sampling implies that d.P n100 where Po is the probability measure under which the random variables Xi, X2,... are independent normals with mean 0 amd variance 1. The...