4)Let X₁, X₂,..., Xn be mutually independent and identically distributed continuousrandom variables with the following density function,f(x) = {630x*(1-x)¹ for 0 < x < 10,otherwisea)Use the Chebychev inequality to find the value of k, for P(0.2 ≤ x ≤ 0.8).b) What is the upper and lower bound of P(0.2 < X < 0.8)?c) Use the central limit theorem to find P(9.5 ≤X ≤ 10.5).

From the given information, fx=630x41-x4, for 0<x<1 Consider, μ=∫01xfxdx =∫01x*630x41-x4dx…

Answered: 4) Let X₁, X₂,..., Xn be mutually…

4) Let X₁, X₂,..., Xn be mutually independent and identically distributed continu random variables with the following density function, for 0

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Question

4)
Let X₁, X₂,..., Xn be mutually independent and identically distributed continuous
random variables with the following density function,
f(x) = {630x*(1-x)¹ for 0 < x < 1
0,
otherwise
a)
Use the Chebychev inequality to find the value of k, for P(0.2 ≤ x ≤ 0.8).
b) What is the upper and lower bound of P(0.2 < X < 0.8)?
c) Use the central limit theorem to find P(9.5 ≤X ≤ 10.5).

Expert Solution

Step 1

From the given information,

$f (x) = 630 x^{4} {(1 - x)}^{4}, f o r 0 < x < 1$

Consider,

$μ = \int_{0}^{1} x f (x) d x = \int_{0}^{1} x * 630 x^{4} {(1 - x)}^{4} d x = \int_{0}^{1} 630 x^{5} {(1 - x)}^{4} d x = 0.5$

$σ^{2} = \int_{0}^{1} x^{2} f (x) d x - μ^{2} = \int_{0}^{1} 630 x^{6} {(1 - x)}^{4} d x - 0 . 5^{2} = 0.2727 - 0.25 = 0.023$

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