1. If X has a gamma distribution with 8 = 4 and a = 2, find P(X < 5) usinga) the integration approachb) the summation approach

Given, has a gamma distribution with anThe pdf is given by,0 \)" data-mce-style="cursor:…

Answered: If X has a gamma distribution with 8 =…

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

1. If X has a gamma distribution with 8 = 4 and a = 2, find P(X < 5) using
a) the integration approach
b) the summation approach

Expert Solution

Step 1: we first define the given terms and solve the first part

Given,

$X$ has a gamma distribution with $theta equals 4$ an $alpha equals space 2$

The pdf is given by,

$f (x) = \frac{1}{Γ (α) θ^{α}} x^{α - 1} e^{\frac{- x}{θ}} . . . w h e r e, x > 0$

$f (x) = \frac{1}{Γ (2) 4^{2}} x^{2 - 1} e^{\frac{- x}{4}}$

$f (x) = \frac{1}{16} x e^{\frac{- x}{4}}$

(a)

The integration approach

$P (x < 5) = \int_{0}^{5} f (x) d x$

$= \int_{0}^{5} \frac{1}{16} x e^{\frac{- x}{4}} d x$

$= \frac{1}{16} \int_{0}^{5} x e^{\frac{- x}{4}} d x$

$equals 1 over 16 open parentheses negative 4 x e to the power of fraction numerator negative x over denominator 4 end fraction end exponent plus 4 integral subscript 0 superscript 5 e to the power of fraction numerator negative x over denominator 4 end fraction end exponent d x close parentheses subscript 0 superscript 5 equals 1 over 16 open parentheses negative 4 x e to the power of fraction numerator negative x over denominator 4 end fraction end exponent minus 16 e to the power of fraction numerator negative x over denominator 4 end fraction end exponent d x close parentheses subscript 0 superscript 5 equals 1 over 16 open parentheses 16 minus left parenthesis 20 plus 16 right parenthesis e to the power of fraction numerator negative 5 over denominator 4 end fraction end exponent close parentheses equals 1 over 16 open parentheses 16 minus 10.3142 close parentheses equals 0.3554$