1. #11 page 363

The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by Yn on the nth day of the year. Finn observes that the differences Xn = Yn+1 − Yn appear to be independent random variables with a common distribution having mean μ = 0 and variance 2 = 1/4. If Y1 = 100, estimate the probability that Y365 is (a) >= 100

mean <- 0
var <- 1/4
sd <- sqrt(var)
n <- 364
y1 <- 100
a <- 100

a1 <- (y1-a)/sqrt(n)
pnorm(a1, mean = mean, sd=sd, lower.tail = FALSE)
## [1] 0.5

  1. = 110

b <- 110
b2 <- (b-y1)/sqrt(n)
pnorm(b2, mean = mean, sd=sd, lower.tail = FALSE)
## [1] 0.1472537

  1. = 120

c <- 120
c3 <- (c-y1)/sqrt(n)
pnorm(c3, mean = mean, sd=sd, lower.tail = FALSE)
## [1] 0.01801584

2. Calculate the expected value and variance of the binomial distribution using the moment generating function.

\[PMF=\left( \begin{matrix} n \\ x \end{matrix} \right) { p }^{ x }{ q }^{ n-x }\] \[MGF=M\left( t \right) =\sum _{ x=1 }^{ n }{ { e }^{ tx } } \left( \begin{matrix} n \\ x \end{matrix} \right) { p }^{ x }{ q }^{ n-x }\] \[=\sum _{ x=1 }^{ n }{ \left( \begin{matrix} n \\ x \end{matrix} \right) } { \left( p{ e }^{ t } \right) }^{ x }{ q }^{ n-x }\] \[={ \left( p{ e }^{ t }+q \right) }^{ n }\] Expected value, take the first dervative of the MGF and sub in t=0 \[=np\] For Variance we do the following: \[=n\left( n-1 \right) { p }^{ 2 }+np\] \[E\left( { X }^{ 2 } \right) -E{ (X) }^{ 2 }=np(1-p)\]

3. Calculate the expected value and variance of the exponential distribution using the moment generating function.

\[f(x)=\lambda { e }^{ -\lambda x }\] \[M`X(t)=\lambda (\lambda -t)2\] \[AND\] \[M``X(t)=2\lambda (\lambda -t)3\]

EXPECTED VALUE:

\[{ M` }_{ x }\left( 0 \right) =\frac { \lambda }{ { \left( \lambda -0 \right) }^{ 2 } } =\frac { \lambda }{ { \lambda }^{ 2 } } =\frac { 1 }{ \lambda } \]

VARIANCE:

\[V\left( X \right) =E\left( { X }^{ 2 } \right) ={ M`` }_{ x }(0)-{ M` }_{ x }(0)=\] \[=\frac { 2\lambda }{ { \left( \lambda -0 \right) }^{ 3 } } -\frac { 1 }{ { \lambda }^{ 2 } } =\frac { 2\lambda }{ { \lambda }^{ 3 } } -\frac { 1 }{ { \lambda }^{ 2 } } =\frac { 2 }{ { \lambda }^{ 2 } } -\frac { 1 }{ { \lambda }^{ 2 } } =\frac { 1 }{ { \lambda }^{ 2 } } \]