Ylu_hw9

1

a>=100

std = sqrt(364/4)
pnorm(0, mean = 0, sd = std, lower.tail = FALSE)

## [1] 0.5

a>=110

std = sqrt(364/4)
pnorm(10, mean = 0, sd = std, lower.tail = FALSE)

## [1] 0.1472537

a>=120

std = sqrt(364/4)
pnorm(20, mean = 0, sd = std, lower.tail = FALSE)

## [1] 0.01801584

2

\(f(x)=C(n,x)p^{x}(1-p)^{n-x}\)

\(M(t)=\sum_{0}^{n}e^{xt}C(n,x)p^{x}(1-p)^{n-x}\)

\(M(t)=[(1-p)+pe^{t}]^{n}\)

\(M'(t)=n(pe^{t})[(1-p)+pe^{t}]^{n-1}\)

\(E= M'(0) = np\)

\(M''(t)=n(n-1)(pe^{t})^{2}[(1-p)+pe^{t}]^{n-2}+n(pe^{t})[(1-p)+pe^{t}]^{n-1}\)

\(M''(0)=n(n-1)p^{2}+np\)

\(var= M''(0)-M'(0)^{2} = np(1-p)\)

3

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

\(M'(x)=\int e^{tx}\lambda e^{-\lambda x}dx = \frac{\lambda}{\lambda - t}\)

$M’(0) = $

\(M''(x)=\frac{2\lambda}{(\lambda - t)^{3}}\)

\(M''(0) = \frac{2}{(\lambda)^{2}}\)

\(var = M''(0)-M'(0)^{2} = \frac{2}{(\lambda)^{2}} - \frac{1}{(\lambda)^{2}}= \frac{1}{(\lambda)^{2}}\)