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

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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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