数学课业代做 MATH 7241代写 数学Problem Set代写 数学代写

MATH 7241: Problem Set #2

数学课业代做 Reading: relevant background material for these problems can be found in the class notes, and in Ross (Chapters 2,3,5) and in Grinstead and Snell

Reading: relevant background material for these problems can be found in the class notes, and in Ross (Chapters 2,3,5) and in Grinstead and Snell (Chapters 1,2 3, 6).

Exercise 1 数学课业代做

A typing firm has three typists A,B and C. The number of errors per 100 pages made by typist A is a Poisson random variable with mean 2.6; the number of errors per 100 pages made by typist B is a Poisson random variable with mean 3; the number of errors per 100 pages made by typist C is a Poisson random variable with mean 3.4. A manuscript of 300 pages is sent to the firm.

Let X denote the number of errors in the typed manuscript.

a) Assume that one typist is randomly selected to do all the work. Find the mean and variance of X.

b) Assume instead that the work is divided into three equal parts which are given to the three typists. Find the mean and variance of X in this case.

Exercise 2 数学课业代做

In a variation of the classic Monty Hall game show, the host sets up five doors and hides prizes behind two of the doors. The contestant first guesses a door, and then the host opens one of the other four doors to show that it does not conceal a prize. The contestant is offered the opportunity to switch her guess to a different door. Should she switch or stay with her original choice? [Hint: see notes on the Monty Hall question, and try to imitate the solution provided there].

Exercise 3 数学课业代做

A fair die is rolled repeatedly. Let X_n be the result of the n^th roll. So X_n takes values {1,…, 6}, each with probability 1/6, and the random variables X₁, X₂,… are all independent. Let

N = min{n : X_n = X_n−1, n ≥ 2}

That is, N is the first roll where the result is equal to the previous roll. [e.g., if you roll the sequence 2, 3, 1, 4, 4, 6,… then N = 5.] Find E[N].

[Hint: argue that E E[N|X₁ = a] = E[N|X₁ = b] for all a, b = 1,…, 6, and conclude that E[N] = [N|a] for any a = 1,…, 6. Then condition on the outcomes of the first two rolls, and imitate the argument we used in class for the ‘rat in a maze’ problem to derive an equation for E[N].]

Exercise 5

Randomly distribute r balls in n boxes so that the sample space consists of n^r equally likely elements. Let N_n be the number of empty boxes. Suppose that r, n → ∞ in such a way that their ratio r/n converges to a constant value c. Show that n⁻¹ E[Nn] converges as n → ∞ and find the limiting value in terms of c.

[Hint: define X_i = 1 if the ith box is empty, and X_i = 0 if the ith box is not empty].

Exercise 6 数学课业代做

Suppose X is an exponential random variable. One of the following three formulas is correct:

(a) E[X² | X > 1] = E[(X + 1)²]

(b) E[X² | X > 1] = E[X²]+1

(c) E[X² | X > 1] = (E[X] + 1)² 

Without doing computations, use the memoryless property of the exponential distribution to explain which answer is correct.

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