The total number of hours, in units of 100 hours,that a family runs a vacuum cleaner over a period ofone year is a random variable X having the densityfunction shown to the right. Find the variance of X.(x - 5),f(x) = 2(x - 5),0,5 The proportion of people who respond to a certain mail-order solicitation is a randomvariable X having the following density function.f(x)=2(x + 3)70,2Find the variance of X.0

The density function of X is given as, The mean for f is,

Answered: The total number of hours, in units of…

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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Related questions

Question

The total number of hours, in units of 100 hours,
that a family runs a vacuum cleaner over a period of
one year is a random variable X having the density
function shown to the right. Find the variance of X.
(x - 5),
f(x) = 2(x - 5),
0,
5<x< 6,
6≤x<7,
elsewhere

The proportion of people who respond to a certain mail-order solicitation is a random
variable X having the following density function.
f(x)
=
2(x + 3)
7
0,
2
Find the variance of X.
0<x< 1,
elsewhere

Expert Solution

Step 1: Determining the mean

The density function of X is given as,

$f left parenthesis x right parenthesis space equals space open curly brackets table row cell left parenthesis x minus 4 right parenthesis comma space 4 less than x less than 5 end cell row cell 2 minus left parenthesis x minus 4 right parenthesis comma space 5 less or equal than x less than 6 end cell row cell 0 comma space e l s e w h e r e end cell end table close$

The mean for f is,

$table row mu equals cell integral subscript space space end subscript superscript space x f left parenthesis x right parenthesis d x end cell row blank equals cell space integral subscript 4 superscript 5 x times left parenthesis x minus 4 right parenthesis d x plus integral subscript 5 superscript 6 x times left parenthesis 2 minus left parenthesis x minus 4 right parenthesis right parenthesis d x end cell row blank equals cell integral subscript 4 superscript 5 x squared minus 4 x space d x space plus space integral subscript 5 superscript 6 x left parenthesis 2 minus x plus 4 right parenthesis d x end cell row blank equals cell integral subscript 4 superscript 5 left parenthesis x squared minus 4 x right parenthesis d x plus integral subscript 5 superscript 6 left parenthesis 6 x minus x squared right parenthesis d x end cell row blank equals cell open square brackets x cubed over 3 close square brackets subscript 4 to the power of 5 space minus 4 open square brackets x squared over 2 close square brackets subscript 4 to the power of 5 plus 6 open square brackets x squared over 2 close square brackets subscript 5 to the power of 6 minus open square brackets x cubed over 3 close square brackets subscript 5 to the power of 6 end cell row blank equals cell open square brackets 5 cubed over 3 minus 4 cubed over 3 close square brackets minus 4 open square brackets 5 squared over 2 minus 4 squared over 2 close square brackets plus 6 open square brackets 6 squared over 2 minus 5 squared over 2 close square brackets minus open square brackets 6 cubed over 3 minus 5 cubed over 3 close square brackets end cell row blank equals cell 20.333 space minus open parentheses 4 cross times 4.5 close parentheses plus open parentheses 6 cross times 5.5 close parentheses minus left parenthesis 30.333 right parenthesis end cell row blank equals 5 end table$