Problem 1

a)
\(E({ S }_{ 365 })\quad =\quad 0\)
\(\Phi (0)\quad =\quad .5\)

b)

\(P({ S }_{ 365 })>10\quad \cong \quad1-\Phi (\frac { 10 }{ \sqrt { 365*.25 } } )\quad\)
\(=1-\Phi (.74)\quad\)

1 - pnorm(10/sqrt(365*.5))
## [1] 0.2295792

c)

\(P({ S }_{ 365 })>20\quad \cong \quad1- \Phi (\frac { 20 }{ \sqrt { 365*.25 } } )\quad\)
\(\quad =\quad1- \Phi (1.48)\quad\)

1 - pnorm(20/sqrt(365*.5))
## [1] 0.06937441

Problem 2

Binomial Distribution MGF: \({ (p{ e }^{ t }+q) }^{ n }\)

1st derivative:

\(\frac { d }{ dt } { (p{ e }^{ t }+q) }^{ n }\)
\(=n{ (p{ e }^{ t }+q) }^{ n-1 }p{ e }^{ t }\)

2nd derivative:

\(\frac { d }{ dt } (n{ (p{ e }^{ t }+q) }^{ n-1 }p{ e }^{ t })\)
\(=\frac { d }{ dt } (n{ (p{ e }^{ t }+q) }^{ n-1 })*p{ e }^{ t }+\frac { d }{ dt } (p{ e }^{ t })*(n{ (p{ e }^{ t }+q) }^{ n-1 }\)
\(=(n-1)n{ (p{ e }^{ t }+q) }^{ n-2 }{ p }^{ 2 }{ e }^{ 2t }+p{ e }^{ t }(n{ (p{ e }^{ t }+q) }^{ n-1 }\)

1st Moment:

\(=n{ (p{ e }^{ 0 }+q) }^{ n-1 }{ p }{ e }^{ 0 }\)
\(=n{ (1) }^{ n-1 }{ p }\)
\(=np\)

2nd Moment

\(=(n-1)n{ (1) }^{ n-2 }{ p }^{ 2 }*{ 1 }+p*1(n{ (1) }^{ n-1 }\)
\(=n(n-1){ p }^{ 2 }+np\)

variance

\(=n(n-1){ p }^{ 2 }+np\quad -\quad { n }^{ 2 }{ p }^{ 2 }\)
\(={ n }^{ 2 }{ p }^{ 2 }-n{ p }^{ 2 }+np\quad -\quad { n }^{ 2 }{ p }^{ 2 }\)
\(=np-\quad n{ p }^{ 2 }\)
\(=np(1-p)\)

Problem 3

Exponential Distribution MGF: \(\quad =\quad \frac { \lambda }{ \lambda -t }\)

1st derivative

\(\frac { d }{ dt } (\frac { \lambda }{ \lambda -t } )\)
\(=\frac { -(-1)\lambda }{ { (\lambda -t) }^{ 2 } }\)
\(=\frac { \lambda }{ { (\lambda -t) }^{ 2 } }\)

2nd derivative

\(=\frac { d }{ dt } \frac { \lambda }{ { (\lambda -t) }^{ 2 } }\)
\(=\frac { -2(\lambda -t)*-1*\lambda }{ { (\lambda -t) }^{ 4 } }\)
\(=\frac { 2\lambda (\lambda -t) }{ { (\lambda -t) }^{ 4 } }\)
\(=\frac { 2\lambda }{ { (\lambda -t) }^{ 3 } }\)

1st moment

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

2nd moment

\(=\frac { 2\lambda }{ { (\lambda -0) }^{ 3 } }\)
\(=\frac { 2 }{ \lambda ^{ 2 } }\)

variance

\(=\frac { 2 }{ \lambda ^{ 2 } } -\frac { 1 }{ { \lambda }^{ 2 } }\)
\(=\frac { 1 }{ { \lambda }^{ 2 } }\)