Let X1, ..., X20 be independent Poisson random variables with mean one. 1 2 (a) Use the Markov inequality to obtain a bound on P (X1 + X2 + · · · + X20 > 15). (b) Use the central limit theorem to approximate the above probability.
Let X1, ..., X20 be independent Poisson random variables with mean one. 1 2 (a) Use...
Let X1, · · · , X20 be independent Poisson random variables with mean (print please) 1. (i) Use the pmf of Poisson distribution to compute P(X1 + · · · + X20 > 15). (ii) Use the Markov inequality to obtain a bound on P(X1 + · · · + X20 > 15). (iii) Use the central limit theorem to approximate P(X1 + · · · + X20 > 15).
Problem 8 (4x4 pts) Suppose Xi, X2-, ..,. Xn are each independent Poisson random variables with mean 1. Let 100 k=1 (a) Rccall, Markov's incquality is P(Y > a) for a> 0 Using Markov's inequality, estimate the probability that P(Y > 120). (b) Rccal, Chebyshev's incquality is Using Chebyshev's inequality, estimate P( Y-?> 20) (c), (d) Using the Central Limit Theorem, estimate P(Y > 120) and Ply-? > 20).
Using Central Limit Theorem) Let S10 sum of 10 Poisson random variables each with mean = 1 1. Find P(S 10 > 10) exactly using Minitab CDF command (Poisson mean =10). 2. Approximate above probability using bell curve approximation -- Normal mean = 0 and standard deviation 1. 3. Show Minitab Command line output
Let X1 and X2 be random variables, not necessarily independent. Show that E [X1 + X2] = E [X1] + E [X2]. You may assume that X1 and X2 are discrete with a joint probability mass function for this problem, while the above inequality is true also for continuous random variables.
Question 3: Let X1,..., X.be iid Poisson (2) random variables. a. Find the maximum likelihood estimate for X. b. Obtain the Fisher expected information. c. Obtain the observed information evaluated at the maximum likelihood estimate. d. For large n, obtain a 95% confidence interval for based on the Central Limit Theorem. e. Repeat part (a), but use the Wald method. f. Repeat part (d), but use the Score method. 8. Repeat part (a), but use the likelihood ratio method.
Let X1, Y.X2, ½, distributed in [0,1], and let ,X16, Y16 be independent random variables, uniformly 2. 16 Find a numerical approximation to P(IW E(W)l< 0.001) HINT: Use the central limit theorem
Let X1, X2, X3 be independent random variables with E(X1) = 1, E(X2) = 2 and E(X3) = 3. Let Y = 3X1 − 2X2 + X3. Find E(Y ), Var(Y ) in the following examples. X1, X2, X3 are Poisson. [Recall that the variance of Poisson(λ) is λ.] X1, X2, X3 are normal, with respective variances σ12 = 1, σ2 = 3, σ32 = 5. Find P(0 ≤ Y ≤ 5). [Recall that any linear combination of independent normal...
1. The random variables Xi, X2,... are independent and identically distributed (iid), . .. are independent and identica each with pdf f given in Assignment 4, Question 1. Let s, X1 + . .. + Xn. 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*. Verify your answer by simulating 10,000 outcomes of S1o0 and counting how many...
3. (25 pts.) Let X1, X2, X3 be independent random variables such that Xi~ Poisson (A), i 1,2,3. Let N = X1 + X2+X3. (a) What is the distribution of N? (b) Find the conditional distribution of (X1, X2, X3) | N. (c) Now let N, X1, X2, X3, be random variables such that N~ Poisson(A), (X1, X2, X3) | N Trinomial(N; pi,p2.ps) where pi+p2+p3 = 1. Find the unconditional distribution of (X1, X2, X3). 3. (25 pts.) Let X1,...
Let X1, X2 ... be a sequence of random variables. Write down the definition of "Xin con- verges to X in distribution". State the central limit theorem formally. Make sure to include all the assumptions of the theorem. Ten people wait in line to pay their items at a store. Suppose the amount of time (in minutes) that each customer takes to pay is a Uniform([1, 3]) random variable. Use the central limit theorem to approximate the probability that it...