Reference no: EM132621717

Question 1. (Markov and Cliehysliev Inequalities)
A statistician wants to estimate the mean height h (in motel's) of a population, based on n independent samples X1,.....Xn, chosen uniformly from the entire population. He uses the sample mean M_n = (X₁ + ... + Xn)/n as the estimate of it, and a rough guess of LO meters for the standard deviation of the samles X_i.

(a) How large should n be so that the standard deviation of M_n is at most 1 Centimeter?

(b) How large should n be so that Chebyshev's inequality guarantees that the estimate is within 5 centimeters from h, with probability at lest 0.99?

(c) The statistician realizes that all persons in the population have heights between 1.4 and 2.0 meters and revises the standard deviation figure that he uses based on the bound of Example 5.3 (pg. 268 in the textbook, the derivation that was emailed out to everybody last week.) How should the values of n obtained in parts (a) and (b) be, revised?

Question 2. (The Weak Law of Large Numbers)

In order to estimate f, the true fraction of smokers in a large population, Alvin selects n people at random. His estimator Mn Is obtained by dividing Sn, the number of smokers in his sample, by N, i.e. , Mn, = Sn/n, Alvin chooses the sample size n to be the smallest possible number for which the Chebyshev inequality yields a guarantee that

P(|M_n - f| ≥ ∈) ≤ δ,

where ∈ and δ are some prespecified tolerances, Determine how the value of n. recommended by the Chebyshev inequality changes in the following cases.

(a) The value of ∈ is reduced to half its original value.
(b) The value of δ is reduced to half its original value.

Question 3. (Convergence in Probability)
Let X1, X2, ... be it random variables that are uniformly distributed over [1,-1]. Show that the sequence Y1, Y2, .... converges in Probability to some limit, and identify the limit for each of the following cases:

(a) Yn = Xn/n
(b) Yn = (Xn)ⁿ
(c) Yn = X1.X2 ... Xn
(d) Yn = max{X1,.....Xn}

Question 4 Before starting to play roulette in a casino, you want to look at biases that you can exploit. You therefore watch 100 rounds that result in a number between 1 and 36, and count the number of rounds for which the result is odd. If the count exceeds 55, you decide that the roulette is not fair. Assuming that the roulette is fair, find an approximation for the probability that you will make the wrong decision.

Question 5 (The Central Limit Theorem)

During each day, the probability that your computer's operating system crashes at least once is 5%, independent of every other day. Your are interested in the probability of at least 45 crash free days out of the next 50 days

(a) Find the probability of interest by using the normal approximation to the binomial

(b) Repeat, this time using the Poisson approximation ro the binomial.

Question 6 (The Central Limit Theorem)

A factory produces Xn gadgets on day n where Xn are independent and identically distributed random variables with mean 5 and variance 9.

(a) Find an approximailun to the probability that the total number of gadgets produced in 100 days is less than 440.

(b) Find (approximately) the largest value of n such that.

P(X₁ +.... +X_N ≥ 200 + 5n) ≤ 0.05

(c) Let N be the first day on which the total number of gadgets produced exeeeds 1000. Calculate an approximation to the probability that N ≥ 220.