The thickness of a wooden shelf in mm, has probability density function f(x) = {0.75 – 0.75 (x – 5)2} : for 4</= x </ =6 and 0 otherwiseFind:the meanσ2The probability that the thickness is less than 5mm

Given probability density functionf(x)=0.75-0.75(x-5)2 4≤x≤60otherwise 1.The mean of the…

Answered: The thickness of a wooden shelf in mm,…

Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018

18th Edition

ISBN:9780079039897

Author:Carter

Publisher:Carter

Chapter10: Statistics

Section10.1: Measures Of Center

Problem 9PPS

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Question

The thickness of a wooden shelf in mm, has probability density function

f(x) = {0.75 – 0.75 (x – 5)²} : for 4</= x </ =6 and 0 otherwise

Find:

the mean
σ²
The probability that the thickness is less than 5mm

Expert Solution

Step 1

Given probability density function
$f (x) = [\begin{matrix} \{0.75 - 0.75 {(x - 5)}^{2}\} & 4 \leq x \leq 6 \\ 0 & o t h e r w i s e \end{matrix}]$

1.
The mean of the probability density function is calculated as shown below
$E (X) = \int_{- \infty}^{\infty} x * f (x) d x = \int_{- \infty}^{4} 0 d x + \int_{4}^{6} x * (0.75 - 0.75 {(x - 5)}^{2}) d x + \int_{6}^{\infty} 0 d x = \int_{4}^{6} (0.75 x - 0.75 x {(x - 5)}^{2}) d x = \int_{4}^{6} (0.75 x - 0.75 x^{3} - 18.75 x + 7.5 x^{2}) d x = \int_{4}^{6} (- 0.75 x^{3} - 18 x + 7.5 x^{2}) d x = {[- \frac{0.75 x^{4}}{4} - \frac{18 x^{2}}{2} + \frac{7.5 x^{3}}{3}]}_{4}^{6} = - \frac{0.75 (6^{4} - 4^{4})}{4} - \frac{18 (6^{2} - 4^{2})}{2} + \frac{7.5 (6^{3} - 4^{3})}{3} = - 195 - 180 + 380 = 5$
The mean of the probability density function is 5.

2.
The variance of the probability distribution is calculated as shown below
$V a r (X) = \int_{- \infty}^{\infty} x^{2} * f (x) d x - μ^{2} = \int_{- \infty}^{4} 0 d x + \int_{4}^{6} x^{2} * (0.75 - 0.75 {(x - 5)}^{2}) d x + \int_{6}^{\infty} 0 d x - 5^{2} = \int_{4}^{6} (0.75 x^{2} - 0.75 x^{2} {(x - 5)}^{2}) d x - 25 = \int_{4}^{6} (0.75 x^{2} - 0.75 x^{4} - 18.75 x^{2} + 7.5 x^{3}) d x - 25 = \int_{4}^{6} (- 0.75 x^{4} - 18 x^{2} + 7.5 x^{3}) d x - 25 = {[- \frac{0.75 x^{5}}{5} - \frac{18 x^{3}}{3} + \frac{7.5 x^{4}}{4}]}_{4}^{6} - 25 = - \frac{0.75 (6^{5} - 4^{5})}{5} - \frac{18 (6^{3} - 4^{3})}{3} + \frac{7.5 (6^{4} - 4^{4})}{4} - 25 = - 1012.80 - 912 + 1950 - 25 = 0.2$
The variance of probability distribution function is 0.2.

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The thickness of a wooden shelf in mm, has probability density function f(x) = {0.75 – 0.75 (x – 5)2} : for 4= x </ =6 and 0 otherwise Find: the mean σ2 The probability that the thickness is less than 5mm