Problem 1

The price of one share of stock in the Pilsdorff Beer Company (see Exercise 8.2.12) is given by \(Y_n\) on the nth day of the year. Finn observes that the differences \(X_n = Y_{n+1} − Y_n\) appear to be independent random variables with a common distribution having mean \(\mu = 0\) and variance \(\sigma^2 = 1/4\). If \(Y_1 = 100\), estimate the probability that \(Y_{365}\) is

  1. ≥ 100.
q = 100 - 100
n = 364
std <- sqrt(1/4)
1 - pnorm(q = q, sd = std)
## [1] 0.5
  1. ≥ 110.
q = (110-100)/sqrt(n)
1 - pnorm(q = q, sd = std)
## [1] 0.1472537
  1. ≥ 120.
q = (120-100)/sqrt(n)
1 - pnorm(q = q, sd = std)
## [1] 0.01801584

Problem 2

Calculate the expected value and variance of the binomial distribution using the moment generating function. Let \(q=1-p\)

\[\begin{align*} g(t) = \sum_{j=0}^{n} e^{tj}{n \choose j}p^jq^{n-j} = \sum_{j=0}^{n} {n \choose j}(pe^t)^jq^{n-j} =(pe^t+ q)^n \end{align*}\]

Expected value: Taking the derivative of the function with respect to t

\[E(X) =u_1=\frac{dg(0)}{dt} = pe^t n(pe^t+ q)^{n-1} = np(p + q)^{n-1} = np (p + 1 - p)^{n-1} = np\] Variance: Taking the second derivative

\[\frac{dg(0)}{dt} - (\frac{d^2g(0)}{dt^2})^2 = np - n(n-1)p^2+np\] \[\sigma^2 = u_2 - u_1^2 = n(n-1)p^2+np - (np)^2 = np(1-p)\]

Problem 3

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

\[g(t) = \sum_{j=1}^{\infty} e^{tx}\lambda e^{-\lambda x} = \frac{\lambda}{t- \lambda}\] Expected value:

\[E(x) = u_1 = \frac{\lambda}{(t- \lambda)^2}|_{t=0} = \frac{1}{\lambda}\]

Variance:

\[ u_2 = \frac{2\lambda}{(\lambda-t)^3}=\frac{2}{\lambda^2}\]

\[V = u_2-u_1^2 = \frac{2\lambda}{(t- \lambda)^3}|_{t=0} = \frac{1}{\lambda^2}\]