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).72-00(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: a) 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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Exercise 11. Suppose that X ~ Expon(5), i.e. f(x) = e-/5 1(x > 0). Compute. Compute... (a) E(X), (b) q0.5, the median, and (c) var(X).
Let f(9) = – 3 and f'(x) > 1 for 9 1, we have f(11) > : this the smallest f(11) can be. %23 a W E R G
3. A random variable X has a cumulative distribution function if 0 0.5). (e) Find P(X < 1.25). (f) Find P(X = 1.25).
(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.
3. Suppose X₁,..., Xn are iid with mean and variance o2 <∞o. Let Xn be the sample mean. (a) State the definition of convergence in distribution of a sequence of random variables. (b) Find the limiting distribution of Xn. (c) What is the limiting distribution of the random variable √n(Xn-u)/o? State the central limit theorem. (7 n)//n. Show
6
a) Suppose that X₁, X₂, X3,...,X, is a random sample of n from population X which has a c square distribution with one degrees of freedom. (1) Find lim Mg(t). 72-00
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…
3) Given the joint distribution f (x1, x2) = a x,X2 for 0 < x, < 1, 0 < x2 < 1, and 0 elsewhere , find a possible value for a first. Then determine one number b randomly in any meaningful way you like. (b + 0) and calculate i)P(bX, +bX, < 1) ii )E(X, | X, = 1) - (E(X¡ | X, = 0.5) %D Make sure to write down all the details that you actually went through while solving this exercise in a very clear manner.
The number of hours per week that the television is turned on is determined for each family in a sample. The mean of the data is 31 hours and the median is 27.2hours. Twenty-four of the families in the sample turned on the television for 16 hours or less for the week. The 8th percentile of the data is 16 hours. Based on the given information, determine if the following statement is true or false. The first quartile is less than 16 hours.

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