Answered: (i) Find the mean of X (ii) Find the…

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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Question

The probability distribution of X is given as:

f(x) = { 0.2; x = 1

(x-1)/2; 1 < x < 2

(3-x)/2; 2 ≤ x < 3

0.3; x = 3

0; elsewhere]

(i) Find the mean of X (ii) Find the variance of X

Expert Solution

Step 1: we define the given probability distribution of X then find out the mean of X

Given,

$f left parenthesis x right parenthesis equals open curly brackets table row cell 0.2 end cell cell x equals 1 end cell row cell fraction numerator left parenthesis x minus 1 right parenthesis over denominator 2 end fraction end cell cell 1 less than x less than 2 end cell row cell fraction numerator left parenthesis 3 minus x right parenthesis over denominator 2 end fraction end cell cell 2 less or equal than x less than 3 end cell row cell 0.3 end cell cell x equals 3 end cell row 0 cell e l s e end cell end table close$

(i)

we calculate the mean $left parenthesis mu right parenthesis$ of $X$

$mu equals sum x. f left parenthesis x right parenthesis w e space h a v e space g i v e n space p r o b a b l i t y space d i s t r i b u t i o n space f left parenthesis x right parenthesis comma l. e f left parenthesis 1 right parenthesis equals 0.2 f left parenthesis x right parenthesis equals fraction numerator left parenthesis x minus 1 right parenthesis over denominator 2 end fraction space f o r space space space space 1 less than x less than 2 f left parenthesis x right parenthesis equals fraction numerator left parenthesis 3 minus x right parenthesis over denominator 2 end fraction space f o r space space space space 2 less or equal than x less than 3 f left parenthesis 3 right parenthesis equals 0.3$

let's calculate the mean $left parenthesis mu right parenthesis$ :

$mu equals 1 asterisk times f left parenthesis 1 right parenthesis plus integral subscript 1 superscript 2 x asterisk times fraction numerator left parenthesis x minus 1 right parenthesis over denominator 2 end fraction d x plus integral subscript 2 superscript 3 x asterisk times fraction numerator left parenthesis 3 minus x right parenthesis over denominator 2 end fraction d x plus 3 asterisk times f left parenthesis 3 right parenthesis space space equals 1 asterisk times 0.2 plus integral subscript 1 superscript 2 x asterisk times fraction numerator left parenthesis x minus 1 right parenthesis over denominator 2 end fraction d x plus integral subscript 2 superscript 3 x asterisk times fraction numerator left parenthesis 3 minus x right parenthesis over denominator 2 end fraction d x plus 3 asterisk times 0.3 space space equals 0.2 plus integral subscript 1 superscript 2 fraction numerator left parenthesis x squared minus x right parenthesis over denominator 2 end fraction d x plus integral subscript 2 superscript 3 fraction numerator left parenthesis 3 x minus x squared right parenthesis over denominator 2 end fraction d x plus 0.9 space space equals 0.2 plus open parentheses open square brackets x cubed over 6 close square brackets subscript 1 superscript 2 minus open square brackets x squared over 4 close square brackets subscript 1 superscript 2 close parentheses plus open parentheses open square brackets fraction numerator 3 x squared over denominator 4 end fraction close square brackets subscript 2 superscript 3 minus open square brackets x cubed over 6 close square brackets subscript 2 superscript 3 close parentheses plus 0.9 space space equals 0.2 plus open parentheses open square brackets fraction numerator 2 cubed minus 1 cubed over denominator 6 end fraction close square brackets minus open square brackets fraction numerator 2 squared minus 1 squared over denominator 4 end fraction close square brackets close parentheses plus open parentheses open square brackets fraction numerator 3 asterisk times 3 squared minus 3 asterisk times 2 squared over denominator 4 end fraction close square brackets minus open square brackets fraction numerator 3 cubed minus 2 cubed over denominator 6 end fraction close square brackets close parentheses plus 0.9 space space equals 0.2 plus open parentheses open square brackets fraction numerator 8 minus 1 over denominator 6 end fraction close square brackets minus open square brackets fraction numerator 4 minus 1 over denominator 4 end fraction close square brackets close parentheses plus open parentheses open square brackets fraction numerator 27 minus 12 over denominator 4 end fraction close square brackets minus open square brackets fraction numerator 27 minus 8 over denominator 6 end fraction close square brackets close parentheses plus 0.9 space space equals 0.2 plus 7 over 6 minus 3 over 4 plus 15 over 4 minus 19 over 6 plus 0.9 space space equals 0.2 minus 12 over 6 plus 12 over 4 plus 0.9 space space equals 0.2 minus 2 plus 3 plus 0.9 mu equals 2.1$

Here we have findout

The $bold italic m bold italic e bold italic a bold italic n bold left parenthesis bold italic mu bold right parenthesis bold equals bold 2 bold. bold 1$

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