Question

Suppose that the probability density function of the length of computer cables is f (x) = 0.1 from 1200 to 1210 millimeters.

Answer 1

Answer:

The mean of the cable length is $\mu=1205$

The standard deviation of the cable length is $\sigma=2.886$

Half of the cables lie in the specifications.

Step-by-step explanation:

Point a:

Suppose X is a continuous random variable with probability density function f(x).

The mean of a continuous random variable, denoted as μ or E(X) is

$\mu= E(X)=\int\limits^\infty_{-\infty} {xf(x)} \, dx$

The standard deviation of X is

$\sigma=\sqrt{\sigma^2}$

where $\sigma^2$ is the variance of X

$\sigma^2=\int\limits^\infty_{-\infty} {x^2f(x)} \, dx-\mu^2$

We know that the probability density function of the length of computer cables is

$f(x)=0.1, \:1200$

Applying the above definition of the mean we get

$E(X)=\int\limits^{1210}_{1200} {0.1x} \, dx =1205\\\\\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx\\\\0.1\cdot \int _{1200}^{1210}xdx\\\\\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1\\\\0.1\left[\frac{x^{1+1}}{1+1}\right]^{1210}_{1200}\\\\Simplify\\\\0.1\left[0.5x^2\right]^{1210}_{1200}\\\\\mathrm{Compute\:the\:boundaries}:\quad \left[0.5x^2\right]^{1210}_{1200}=12050\\\\0.1\cdot \:12050=1205$

Applying the above definition of the standard deviation we get

First we need to calculate the variance of X

$\sigma^2=\int\limits^{1210}_{1200} {x^2\cdot 0.1} \, dx-\mu^2$

$\int _{1200}^{1210}0.1x^2dx\\\\\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx\\\\0.1\cdot \int _{1200}^{1210}x^2dx\\\\\mathrm{Apply\:the\:Power\:Rule}:\quad \int x^adx=\frac{x^{a+1}}{a+1},\:\quad \:a\ne -1\\\\0.1\left[\frac{x^{2+1}}{2+1}\right]^{1210}_{1200}\\\\0.1\left[\frac{1}{3}\cdot x^3\right]^{1210}_{1200}\\\\\mathrm{Compute\:the\:boundaries}:\quad \left[\frac{1}{3}\cdot x^3\right]^{1210}_{1200}=14520333.33\\\\0.1\cdot \:14520333.33=1452033.33$

$\sigma^2=1452033.33-1205^2=8.33$

$\sigma=\sqrt{\sigma^2}=\sqrt{8.33}=2.886$

Point b:

To find what proportion of cables is within specifications you need to:

$P(1195$