a) Suppose the reaction temperature X (in \deg C) in acertain chemical process has a uniform distribution withA = -5 and B = 5. Compute P(X < 0) b) Related toQuestion 1, Compute P(-2 <= X < = 3). c)Relatedto Question 1, For k satisfying-5 < k < k+ 4 < 5,compute P(k < x < k+4)..

Assume that the reaction temperature X (in\degC) in a certain chemical process has a uniform…

Answered: a) Suppose the reaction temperature X…

MATLAB: An Introduction with Applications

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4. Lexicographic preference >, is defined as (æ4, y^) >L (xB,yB) if either x4 > xB or x4 = xB and y4 > yB. (a) What is an indifference curve for Lexicographic >L? (b) Illustrate that Lexicographic preference >L is not continuous.
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).
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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).
A pump operates 1000 hours/year. Under a minimal repair concept, the pump failures generated a non-homogenous Poisson process having the following intensity function with t measured in operating hours. Row(t)=0.00003t^2. a) From the information given, is the rate of occurance of failure (ROCOF) increasing, decreasing or remaining constant? b) Calculate the number of expected failures of the pump over 1000 hours of operation. c) Calculate the MTBF for the 1000-hour operation. d) The repair time of the pump is best described by the following probability density function h(t)=t^2/3 for 0<_t<_3 hours. WHat is the mean time of repair, in hours? e)What is the inherent availability of the pump over the 1000 hours?
50. Let X₁, X2, ..., X50 be i.i.d. with E(X) = 25 and V(X₂) = 10. Consider the random variable defined by 50 X₁ the sample average X = Σ Apply the Central Limit Theorem to answer the following: n i=1 (a) What is the approximate distribution of X (include parameter(s))? (b) Approximate P(X> 25.5). (c) Approximate P(|X-25 <0.8). (d) Approximate the 0.975 quantile (97.5th%tile) of X.
Suppose that a random sample of sizen is taken from a Poisson distribution for which the value of the mean e is unknown, and the prior distribution of e is a gamma distribution for which the mean is Po. Show that the mean of the posterior distribution of e will be a weighted average having the form Y,X, + (1– Yn)Ho, and show that yn →1 as n- *.
If we assume that the distribution of means is normal, then 95% of all sample means a) will fall in the interval m ± 1.96(σm) b) will fall in the interval m ± 1.645(σm) c) will be no more than 1.645(σm) from μ d) will be no more than from 1.96(σm) from μ
4. The distribution function of a random variable W is given by: Oforw 2 a. Find P (1 < w< 3). Enter probability as a reduced fraction. prob = %3D b. Find f(w). Enter coefficients as reduced fractions and express powers with ^. f(w) 3= for -2< w < 2 %3D

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a) Suppose the reaction temperature X (in \deg C) in a certain chemical process has a uniform distribution with A = -5 and B = 5. Compute P(X < 0) b) Related to Question 1, Compute P(-2 < = x <= 3). c)Related to Question 1, For k satisfying-5