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,…

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...

See similar textbooks

Related questions

Concept explainers

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

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

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.

Step by step

Solved in 2 steps

Knowledge Booster

$Continuous Probability Distribution$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, probability and related others by exploring similar questions and additional content below.

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