Probability

Question 1

Question 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 σ 2 = 1/4.

Central Limit Theorem

Let \[ S_n = X_1 + X_2 + ... + X_n \] be the sum of n independent continuous random variables with common density function p having expected value µ and variance σ^2. Let

\[S_n^* = (S_n - n\mu)/\sqrt{n}\sigma\]

for all a < b: \[\lim_{n\to\infty} P(a < S^*_n < b) = \frac{1}{\sqrt{2\pi}} \int^b_a e^{\frac{-x^2}{2}}dx\]

If Y1 = 100, estimate the probability that Y365 is

(a) ≥ 100.

\[S_n^* = (S_n - S_n1 - n\mu)/\sqrt{n}\sigma\]

\[S_n^* = (100 - 100 - (365-1)*0)/\frac{\sqrt{365-1}}{2}\]

\[S_n^* = 0\]

## [1] 0.5

(b) ≥ 110.

\[S_n^* = (S_n - S_n1 - n\mu)/\sqrt{n}\sigma\]

\[S_n^* = (100 - 110 - (365-1)*0)/\frac{\sqrt{365-1}}{2}\]

\[S_n^* = -1.048\]

## [1] 0.1473193

(c) ≥ 120.

\[S_n^* = (S_n - S_n1 - n\mu)/\sqrt{n}\sigma\]

\[S_n^* = (100 - 120 - (365-1)*0)/\frac{\sqrt{365-1}}{2}\]

\[S_n^* = -2.0966\]

## [1] 0.0180145

Question 2

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

Binomial distribution moment generating function:

\[M_X(t)= \frac{\lambda}{\lambda - t}\]

E[X] (first derivative of moment generating function):

\[\frac{dM_X(t)}{dt} = \frac{\lambda}{(\lambda-t)^2}\]

E[X^2] (first derivative of moment generating function):

\[\frac{d^2M_X(t)}{dt^2} = \frac{2}{\lambda^2}\] Variance = E[X^2] - E[X]^2

\[Var = \frac{2}{\lambda^2} - \frac{\lambda^2}{(\lambda-t)^4}\]

Question 3

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

Exponential distribution moment generating function:

\[M_X(t)= (pe^t + q)^n\]

E[X] (first derivative of moment generating function):

\[\frac{dM_X(t)}{dt} = np\]

E[X^2] (first derivative of moment generating function):

\[\frac{d^2M_X(t)}{dt^2} = n(n-1)p^2 + np\]

Variance = E[X^2] - E[X]^2

\[Var = n(n-1)p^2 + np - n^2p^2 = p^2(n^2-n-n^2) + np = -np^2 + np\]