PS1. 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 ยต = 0 and variance \(\sigma^2 = 1/4\). If \(Y_1 = 100\), estimate the probability that \(Y_{365}\) is
pnorm(100-100, mean = 0, sd = sqrt(364/4), lower.tail = F)## [1] 0.5pnorm(110-100, mean = 0, sd = sqrt(364/4), lower.tail = F)## [1] 0.1472537pnorm(120-100, mean = 0, sd = sqrt(364/4), lower.tail = F)## [1] 0.01801584PS2: Calculate the expected value and variance of the binomial distribution using the moment generating function.
The moment generating function:
\(M(t)=[(1-p) +pe^t]^n\).
\(M'(t) = n(pe^t)[(1-p)+pe^t]^{n-1}\)
Expected value: \(M'(0) = np\)
\(M''(t) = n(n-p)(pe^t)^2[(1-p)+pet]^{n-2} + n(pe^t)[(1-p)+pe^t]^{n-1}\)
Variance: \(M''(0) = n(n-1)p^2 + np\)
\(\sigma^2 = M''(0) - [M'(0)]^2 = np(1-p)\)
PS3: Calculate the expected value and variance of the exponential distribution using the moment generating function.
\(M(t) = \frac{\lambda}{\lambda-t}\)
\(M'(t) = \frac{\lambda}{(\lambda-t)^2}\)
Expected Value: \(M'(0) = 1/{\lambda}\)
\(M''(t) = \frac{2\lambda}{(\lambda-t)^3}\)
\(M''(0) = \frac2{(\lambda)^2}\)
\(\sigma^2 = \frac2{(\lambda)^2} - (1/{\lambda})^2 = (1/{\lambda})^2\)