 Solution

ID: past-exam-of-the-probability-statistics-and-options-i-course-of-the-university-of-oxford/2009/m/3/c/solution

W_{i}

have mean two and variance four by part (b). Their standardization is

\frac{\sum _{i} W _{i} - 2 n}{2 n} = \frac{1}{2 n} \sum_{i} W_{i} - n .

The central limit theorem therefore proves the claimed convergence to the standard normal law, whose continuous cumulative distribution function is

(2 π)^{- 1/2} \int_{- \infty}^{x} e^{- u^{2} /2} d u

at every real

x

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