Assignment 9

  1. The price of one share of stock in the Pilsdorff Beer Company (see Exer- cise 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 σ2 = 1/4. If Y1 = 100, estimate the probability that Y365 is
  1. ≥ 100.
pnorm(100-100, mean = 0, sd = sqrt(364/4), lower.tail = F)
## [1] 0.5
  1. ≥ 110.
pnorm(110-100, mean = 0, sd = sqrt(364/4), lower.tail = F)
## [1] 0.1472537
  1. ≥ 120.
pnorm(120-100, mean = 0, sd = sqrt(364/4), lower.tail = F)
## [1] 0.01801584
  1. Calculate the expected value and variance of the binomial distribution using the moment generating function.

The moment generating function: M(t)=[(1−p)+pet]n M′(t)=n(pet)[(1−p)+pet]n−1

Expected value: M′(0)=np M″(t)=n(n−p)(pet)2[(1−p)+pet]n−2+n(pet)[(1−p)+pet]n−1

Variance: M″(0)=n(n−1)p2+np σ2=M″(0)−[M′(0)]2=np(1−p)

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

M(t)=λλ−t M′(t)=λ(λ−t)2

Expected Value: M′(0)=1/λ M″(t)=2λ(λ−t)3 M″(0)=2(λ)2 σ2=2(λ)2−(1/λ)2=(1/λ)2