J_herman_assignment_8

11

• \(N=100\)
• \(mean =1000 =\frac{1}{\lambda}\)

• \(\lambda = \frac{1}{\1000}\)

• exponential distribution = \(\lambda e^{-\lambda x}\)

• question 10 states that the mean of an exponential distribution is equal to mean/n
  • therefore 1000/100=10 hours

14

link

• \(x,y,\lambda......z=x-y\)
• \(f(z) = \int_{-\infty}^{\infty} f_x(x)f_{-y}(z-x)dx \ \)

• \(f_{-y}(z-x) = f_{y}(x-z)\)

• As both equations are exponential equations we know that

\[f_x(x) = \left\{ \begin{array}{ll} 0 & \quad x < 0 \\ \lambda e^{-\lambda x} & \quad x \geq 0 \end{array} \right.\]

• Therefore \[f(z) = \int_0^{\infty} e^{-\lambda x} \lambda e^{-\lambda (z-x)}dx \ \]

• Simplified

\[f(z) = \int_0^{\infty} \lambda e^{-\lambda x}* \lambda e^{-\lambda x}*\lambda e^{\lambda z}dx \ \]

• Pull out constant

\[f(z) =\lambda e^{\lambda z}\int_0^{\infty}\lambda e^{-2\lambda x }dx \]

• Integrate \[f(z) =\lambda e^{\lambda z}(-\frac{1}{2} e^{-2\lambda x })\]

• evaluate \[ 0-- \frac{\lambda}{2}e^{\lambda z} .... z<0 \]…

\[\frac{\lambda}{2}e^{\lambda z} .... z<0 \]

• Being that the distribution of \[ Z = x-y & -Z= -x+y \] \[ f_z(z) = f_z(-z) \] \[ z\geq 0 ... \frac{\lambda}{2}e^{-\lambda z} \]

\[f(z) =\frac{\lambda}{2}e^{\lambda [z]}\]

8.1

\(P(|x-m|\geq a) \leq \frac{Var(x)}{a^2}\)

\(mean=10\)

\(variance= 100/3\)

\(\sigma = \sqrt{100/3}\)

\(P(|X - \mu |\geq k\sigma )\leq 1/k^2\)

\(\epsilon = k \ \sigma ... k= \epsilon / \sigma\)

Create fucntion to solve problem

epsilon <- 2
variance=100/3
my_std <- sqrt(variance)

get_prob <- function(epsilon,my_std,title=NULL){
    k <- epsilon/my_std
    paste((1/k^2),title)
}

variance/(epsilon**2)

## [1] 8.333333

get_prob(epsilon=2,my_std,title= 'at epsilon of 2')

## [1] "8.33333333333334 at epsilon of 2"

get_prob(epsilon=5,my_std, title= 'at epsilon of 5')

## [1] "1.33333333333333 at epsilon of 5"

get_prob(epsilon=9,my_std, title= 'at epsilon of 9')

## [1] "0.411522633744856 at epsilon of 9"

get_prob(epsilon=20,my_std, title= 'at epsilon of 20')

## [1] "0.0833333333333333 at epsilon of 20"