let X be a random variable following an exponentialdistribution with parameter 1³0 with the densityfunction f(x) = 2 e²^x, for x>0 f(x) = 0, elsewhere.(a) How do you derive the distribution Y = 10-e-^x ?(b) How do you derive the moment generating function of y(c) If X₁, X₂ X3...... X₁. are random sample from theprevious exponential distribution with parameter 20(X;'s (i = 1, 2,...,10 ) are independent). Flow do you findthe density function of Z= min{X₁, X₂,P(Z > 2) = PIX₁ > 2,.... X₁₁ > 2)X₁0 3.サ'21''' L

Given that the random variable X follows an exponential distribution with parameter with the…

Answered: let X be a random variable following an…

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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let x be a random variable with moment generating function Mx(t)=(0.6 + 0.4e^t)^20 then the variance of x is
Let X be a continuous random variable whose moment generating function is 4x(t) = (5) ³. Find Var (X)
Let Y be a random variable with pdf f(y)= (1/18)(y^3−1) for 1<y<3.(a) Calculate P (Y > 2.5).(b) Calculate the variance of Y .
The pdf is given as follows: f(x) = { (1/6)e^(1/6x) ; x > 0 0; ew] (i) Find the mean and variance of X using the pdf. (ii) Find the moment generating function of X. (iii) Find the mean and variance of X using the moment generating function.
Let X be a Poisson random variable with parameter k. (a) Show that the moment generating function for X is given by ?? (?) = ?^(?(?^?−1)) (b) Show that E(X) = k. (c) Show that Var (X) = k.
Question 2 Let X be a random variable with moment generating function M(t), and let Y= ax+b. Prove that the moment generating function of Y has this form: M(b+at) e^(bt)xM(at) e^(bt)+M(at) e^(at)xM(bt)
(b) (i) Briefly explain how moment generating function of a random variable, y may be used to Generate its moments. (ii) Let the random variable, y have probability density function S(t) = kye-1/2y y20 , elsewhere Where k> 0 is a constant. Find the moment generating function of y and use it to evaluate the mean and variance of y leaving your results in terms of k.
3. (a) Let X be a random variable with PDF f(x) = 1 |x|, |x| ≤ 1. Obtain the MGF of X. (b) Find the PDF of a random variable Y with MGF for t0, and My (0) = 1. 2 et My(0) = (^=¹)'.
Suppose that X is a random variable for which the moment generating function is as follows: MX(t) = e^(2t^2+3t) for −∞ < t < ∞. Find the mean and variance of X. (b) Suppose that X has moment generating function MX(t) =(3e^t/4 + 1/4)^6(i) Find the p.m.f. of X. (ii) Find the mean and variance of X. (c) A person with some finite number of keys wants to open a door. He tries the keys one-by-one independently at random with replacement. How many trails you expect, from him, to open the door? (d) Obtain the form of moment generating function (m.g.f.) for the following p.m.f. – p(x) = ((2^x)(e^-2))/×!, x = 0,1,2, … . Also calculate the mean and variance from m.g.f.
At the end of summer, the total weight of seeds accumulated by a nest of seed-gathering ants will vary from nest to nest. If the total weight of seeds accumulated by a nest is exponentially distributed with parameter λ = 1/5, (a) What is the probability that the total combined weight of the seeds gathered by 100 nests will be larger than 4 95 pounds by the end of next summer? (b) What is the probability that the average weight of the seeds gathered by the 100 nests is larger than 5.3 ? (c) What assumptions are you making to answer parts (a) and (b)? Do you think those assumptions make sense in the context of this problem? Explain. (d) Tell us about the possibility that the total combined weight of the seeds gathered by 2 nests follows a normal distribution. Justify your answer in the speciﬁc context of this problem (ants, nests, seed-gathering). That is, do not just make generic statements that you think apply to every context in the world.
Q5 Find the variance for the PDF px(x) = e-«/2, x > 0.
The moment generating function of the random variable X is given by mX(s) = e2e^(t)−2 and the moment generating function of the random variable Y is mY (s) =(3/4et +1/4)10. If it is assumed that the random variables X and Y are independent, findthe following:(a) E(XY)(b) E[(X − Y )2](c) Var(2X − 3Y)

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