7.2 Problem 11

\(P({ X }_{ 1 },{ X }_{ 2 }...{ X }_{ n }<x)=1-P({ { X }_{ 1 },{ X }_{ 2 }...{ X }_{ n }>x)\qquad }\)

\(=1-{ (1-F(x)) }^{ n }\)

\(=1-{ e }^{ -1000x/\lambda }\)

\(f(x)=\frac { d }{ dx } 1-{ e }^{ -1000x/\lambda }\)

\(f(x)=\frac { 1000 }{ \lambda } { e }^{ -1000x/lambda }\)

\(E(x)=\int _{ 0 }^{ \infty }{ \frac { 1000 }{ \lambda } x{ e }^{ -1000x/\lambda } }\)

\(u=x, dv = { e }^{-x/10 }\)

\(E(x)=\quad 1/10(-10x{ e }^{ -x/10 }{ | }_{ 0 }^{ \infty }-\int _{ 0 }^{ \infty }{10 { e }^{ -x/10 } }\)

\(E(x)=(1/10)(100)\)

\(=10\)

bulbs <- rexp(10000000, .001)
dim(bulbs) <- c(100000, 100)
mins <- apply(bulbs, 1, min)
mean(mins)
## [1] 9.936558

7.2 Problem 14

8.2 Problem 1

a) P(|X - 10| >= 2)

(100/3)/2**2
## [1] 8.333333

p <= 1

b) P(|X - 10| >= 5)

(100/3)/5**2
## [1] 1.333333

p <= 1

c) P(|X - 10| >= 9)

(100/3)/9**2
## [1] 0.4115226

p <= 0.4115226

d) P(|X - 10| >= 20)

(100/3)/20**2
## [1] 0.08333333

p <= 0.0833333