Discussion9

Continuous Independent Trials

Prob 11 Page 363

The price of one share of stock in the Pilsdorff Beer Company is given by \(Y_{n}\) Finn notices that \(X_{n} = Y_{n+1} - Y_{n}\) appear to be independent random variables with \(\mu = 0\) and \(\sigma =\frac{1}{2}\)

If \(Y_{1} = 100\), estimate the probability that:

1. \(Y_{365}\geq100\)

2. \(Y_{365}\geq110\)

3. \(Y_{365}\geq120\)

Solutions:

First of all, we calculate the expected value and variance for \(Y_{365}\)

\(E(X_{n}) = E(Y_{n+1}-Y_{n}) = E(Y_{n+1})-E(Y_{n})\)

\(E(X_{i}) = 0 \implies E(Y_{n+1})=E(Y_{n})\)

\(E(Y_{1}) = E(100) = 100 \implies E(Y_{365}) = 100\)

$V(Y_{365}) = V(Y_{364} + X_{365}) = V(Y_{364}) + V(X_{365}) = V(Y_{364}) + $

\(= V(Y_{363}) + 2\sigma = ... = 365\sigma = \frac{365}{4}\)

Since \(E(Y_{365}) = 100 \implies P(Y_{365} \geq 100) = 0.5\)

\(\frac{110 - 100}{\sqrt{\frac{365}{4}}} = 1.04685\)

pnorm(-1.04685)

## [1] 0.1475844

\(\implies P(Y_{365}\geq 100) = 0.1476\)

\(\frac{120 - 100}{\sqrt{\frac{365}{4}}} = 2.0937\)

pnorm(-2.0937)

## [1] 0.01814336

\(\implies P(Y_{365}\geq 100) = 0.01814\)