• Problem 3: Find the expected value of the function g(X)=X'where X is a random variabledefined by the density f.(x) (0.5) u(x) ew2)

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Answered: • Problem 3: Find the expected value of…

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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part C D
(b) The distribution of claim size, X (in thousands of pounds), for a certain portfolio is modelled by the following probability density function f(x) = 1½ (ex/2 + xe x/2) x≥0. (i) Find the moment generating function for the claim size X. (ii) Obtain the mean, α2, variance and α3 for the claim size X. (iii) (iv) (v) The risk of claims for this portfolio is said to be homogenous with an average number of claims per year of 75. Find the mean, variance and skewness coefficient of the total annual claim size. The company applies a 6% safety loading for the policies in the portfolio. Estimate the initial reserve for the portfolio that would ensure that the total claims S for one year are covered with a probability of 99.95%, using skewness correction. If the company starts with an initial reserve of £125,000 find out the probability of meeting all claims in one year, using a skewness correction.
Question 6 with 3 parts
If P(A) = 0.52, P(B) = 0.45 and P(A u B) = 0.76, then P(B | A) = (Enter a number between 0 and 1, using two decimal points)
B3. A device has two components, and the lifetimes of these components are modelled by random variables Y₁ and Y₂. The first component cannot fail before the second component fails, and the joint density of Y₁ and Y₂ is determined to be f(y₁, y2) = 2e-¹e-92 for 0 0.) I'(a)
b) The random variable X, has probahility density function (nents 1>0 fa(2) = otherwise Using the moment generating functions, show that (i) X, has a limiting distribution which is degenerate at x = 8 (ii) Z, = n(X, - 8) has a limiting distribution which is exponential with mean 1. %3!
34. Consider the continuous random variable X whose pdf is given by f(r) = (a-r³)Io<<1(1). (a) Find the value of a that makes f(z) a pdf. (b) Find E(X). (c) Find V(X). (d) Find P(X ≤).
5. Suppose X and Y are continuous random variables with joint probability density function (pdf) 32 y fxx(x, y) = { 3² (xy)¹³, if 0 ≤a susi 9 otherwise What are the marginal pdfs X and Y? (3)
3) Find the mean and variance of the random variable X with probability function or density f(x). (a)f(x)={k(1-x), –15xs1 O otherwise (b) ƒ(x) = k/3× , x = 1,2,3,... Note: in this problem, you need to first find the values of k.
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(b) Let X1, X2, ...,X, be a random sample from the pdf Be z for z 2 0 f(z,0) = 0, otherwise (1) Show that the statistic T(X1, X2, . . , Xn) = X, is complete sufficient for 8. %3D i=1

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• Problem 3: Find the expected value of the function g(X)= X'where defined by the density f.(x) (0.5) u(x) ew2) is a random variable