HW 8

Lightbulbs

\[E[X=1] = \frac{1} {\frac{100}{1000}}\]
\[E[X=1] = 10 hrs\]

Density Proof

\[Joint Density of X_1 and X_2 = { \lambda }^{ 2 }{ e }^{ -\lambda ({ x }_{ 1 }+{ x }_{ 2 }) }\]
\[z>0\quad =\quad \int _{ 0 }^{ \infty }{ { \lambda }^{ 2 }{ e }^{ -\lambda z }dx } =\frac { 1 }{ 2 } \lambda { e }^{ -\lambda z } \] \[z<0\quad =\quad \int _{ -z }^{ \infty }{ { \lambda }^{ 2 }{ e }^{ \lambda z }dx }=\frac { 1 }{ 2 } \lambda { e }^{ \lambda z } \]
\[f(z)=\frac { 1 }{ 2 } \lambda { e }^{ \left| \lambda \right| z }\]

Chebyshev’s Inequality

a

k <- (2/sqrt(100/3))
paste("Upper bound is",round(1/k^2,2)) 
## [1] "Upper bound is 8.33"

b

k <- (5/sqrt(100/3))
paste("Upper bound is",round(1/k^2,2)) 
## [1] "Upper bound is 1.33"

c

k <- (9/sqrt(100/3))
paste("Upper bound is",round(1/k^2,2)) 
## [1] "Upper bound is 0.41"

c

k <- (20/sqrt(100/3))
paste("Upper bound is",round(1/k^2,2)) 
## [1] "Upper bound is 0.08"