DATA605 Homework 9

1.

  1. 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 \(X_n = Y_{n+1} - Y_n\) appear to be independent random variables with a common distribution having mean \(\mu = 0\) and variance \(\sigma^2 = \frac{1}{4}\). If \(Y_1 = 100\), estimate the probability that \(Y_365\) is
  1. \(\geq 100 = 0.5\)
1 - pnorm(100, 100, 91)
## [1] 0.5
  1. \(\geq 110 = 0.4562\)
1- pnorm(110, 100, 91)
## [1] 0.4562483
  1. \(\geq 120 = 0.413\)
1 - pnorm(120, 100, 91)
## [1] 0.4130212

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

MGF: \(M_X(t) = (q + pe^t)^n\)
1st moment: \(M^{'}_{X}(t) = n(q+pe^t)^{n+1}pe^t\)
2nd moment: \(M^{''}_{X}(t) = n(n-1)(q+pe^t)^{n-2}p^2e^{2t}+n(q+pe^t)^{n-1}pe^t\)
Expected value (1st moment, t = 0): \(E(X) = M^{'}_{X}(0) = np\)
Expected value (2nd moment, t = 0): \(E(X^2) = M^{''}_{X}(0) = n(n-1)p^2 + np\)
Variance: \(V(X) = E(X^2)-E(X)^2 = n(n-1)p^2 + np - n^2p^2 = npq\)

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

MGF: \(M_X(t) = \frac{\lambda}{\lambda - t}, t < \lambda\)
1st moment: \(M^{'}_{X}(t) = \frac{\lambda}{(\lambda - t)^2}\)
2nd moment: \(M^{''}_{X}(t) = \frac{2\lambda}{(\lambda - t)^3}\)
Expected value (1st moment, t = 0): \(E(X) = M^{'}_{X}(0) = \frac{1}{\lambda}\)
Expected value (2nd moment, t = 0): \(E(X^2) = M^{''}_{X}(0) = \frac{2\lambda}{(\lambda - 0)^3} = \frac{2}{\lambda^2}\)
Variance: \(V(X) = E(X^2)-E(X)^2 = \frac{2}{\lambda^2} - \frac{1}{\lambda^2} = \frac{1}{\lambda^2}\)