HW 9

Stock Price

a) \[{ Y }_{ 365 }\ge 100\]

pnorm(100-100, mean = 0, sd = sqrt(364/4), lower.tail = F)
## [1] 0.5

b) \[{ Y }_{ 365 }\ge 110\]

pnorm(110-100, mean = 0, sd = sqrt(364/4), lower.tail = F)
## [1] 0.1472537

c) \[{ Y }_{ 365 }\ge 120\]

pnorm(120-100, mean = 0, sd = sqrt(364/4), lower.tail = F)
## [1] 0.01801584

Binomial Dist

\[ MGF:{ M }(t)=[(1−p)+pe^t]^n \] \[M'(t) = n(pe^t)[(1−p)+pe^t]^{n−1}\] \[ Expected Value: M'(0)=np\] \[ M″(t)=n(n−1)(pe^t)^2[(1−p)+pet]^{n−2}+n(pe^t)[(1−p)+pe^t]^{n−1}\] \[Varience: M″(0)=n(n−1)p^ 2+np \] \[{ \sigma }^{ 2 }=M″(0)−[M′(0)]^ 2=np(1−p)\]

Exp Dist

\[M(t) = \frac { \lambda }{ \lambda -t } \]
\[M'(t) = \frac { \lambda }{ (\lambda -t)^2 } \]
\[Expected Value = M'(0) = \frac { 1 }{ \lambda } \]
\[M''(t) = \frac { 2\lambda }{ (\lambda -t)^{ 3 } } \]
\[M''(0) = \frac { 2 }{ (\lambda )^{ 2 } } \]
\[{ \sigma }^{ 2 }=\frac { 2 }{ (\lambda )^{ 2 } } -(\frac { 1 }{ \lambda } )^{ 2 }=\frac { 1 }{ \lambda ^{ 2 } } \]