a)Suppose that X₁, X2, X3....,X, is a random sample of n from population X which has a chi-square distribution with one degrees of freedom.(i) Find lim Mx(t).11-0(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.

Given Xi~chi square distribution with one degree of freedom, and they are random sample…

Answered: Suppose that X₁, X2, X3.....X is a…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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A clamped spline S for a function f is defined on [1, 3] byS(x) = S0(x) = 1 + 2(x − 1) + 3(x − 1)^2 − (x − 1)^3, 1 ≤ x ≤ 2,S(x) = S1(x) = a + b(x − 2) + c(x − 2)^2 + d(x − 2)^3, 2 ≤ x ≤ 3.Given f'(1) = 2f'(3), find a, b, c, and d.
5. Suppose that a stick of length 1 is broked at a point U uniformly distributed on (0,1). Let L be the length of the longer of the two resulting pieces. 0 (a) Find P (L>). (b) 1 Find the cumulative distribution function of L.
(i) Let X ~ Poisson(A). Then, using Chebyshev's inequality, show that P(X > 21) < 1/A. (ii) Suppose that the number of errors per computer program has a Pois- son distribution with mean 5. We get 125 programs. Let X1, X2, ..., X125 be the number of errors in the programs. Then, using the central limit theorem, find an approximate value for P(Xn < 5.5).
A.2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter lambda. a) If X1, X2. ..., Xp are the times, in minutes, between Successive customers selected randomly, estimate the parameter of the distribution. b) The randomly selected 15 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, 1.8, 0.9, 1.5 and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c) In order to estimate the distribution parameter with 0.4 error and 4% risk, find the minimum sample size.
3A) Given P=822-30Q2 and MC=12+9Q find the consumer surplus (CS) (Hint: TR to MR and optimum quantity is MR=MC). B) S=200000 e5find the growth rate of S. C) MC=24eQ and FC=72 Find the total cost. D) The probability of waiting in bus station is given with the frequency distribution f(t)= 1/16 t3 find the probability that your waiting time will be between 1 and 2 minutes?
6
Suppose that the claim size distribution of an insurance portfolio follows a Pareto distribution of the form α+1 α f(x) β = B\B+x (i) Derive a formula for the rth moment, ar, of this Pareto distribution in terms of its (r-1) th moment, αr-1. Show your steps clearly with reasons. (ii) From this expression find α3 and α4 using the known result for the mean μ of the above Pareto distribution. (You may assume that α>r, where a is one of the two parameters of the Pareto distribution).
2.0.3 Problem 3: Central Limit Theorem Here we will verify the Central Limit Theorem and reproduce plots similar to those from Wikipedia (https://en.wikipedia.org/wiki/Central_limit_theorem#/media/File:Dice_sum_central_limit_theorer a) Write a function that returns n integer random numbers, uniformly distributed between 1 and 6, inclusively. This represents n throws of a fair 6-sided die. The value that comes up at each throw will be called the "score". b) Generate a distribution of 1000 dice throws and plot it as a histogram normalized to unit area. Compute the mean ₁ and standard deviation o₁ of this distribution. Compare your numerical result to the analytical calculation. c) Generate 1000 sets of throws of N = 2, 3, 4, 5, 10, 20, 30 dice, computing the total sum of dice scores for each set. For each value of N, plot the distribution of total scores, and compute the mean and standard deviation on of each distribution. This should be similar to the plot at the link above. d) Plot the…
20. Let X1, X2, ables, and set X, be independent, Exp(a)-distributed random vari- Y1 = X(1) and Y = X(k) – X(k-1); for 2
7. In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Pois- son distribution with probability mass function f(x), where f(x) = 0" e- /x! and 0 is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was observed that ng quadrats had x bacteria (x = 0,1, ., c) and that m quadrats had more than c bacteria. Show that the maximum likelihood estimate of 0 is given by -1 mf(c) 1- F(c)] = s In – m - where s = Ecn, and F(c) =E{(x). x=1 x=0
7. (12) Show that the hypotheses of the Mean Value theorem are satisfied and find all values satisfying the conclusion for 3 - 7? if r 1
5.7. Let the random variable Y, have a distribution that is b(n, p). (a) Prove that Y,/n converges in probability to p. This result is one form of the weak law of large numbers. (b) Prove that 1 Y/n converges in probability to 1-p.

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