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Lecture 12 - Recall some basic statistics The normal...

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Recall some basic statistics: The normal distribution is the bell-shaped distribution that is often used in statistics. It is useful because the (properly scaled) mean of independent random draws of many other statistical distributions will tend toward a normal distribution this is the Central Limit Theorem. Some basic facts and notation: a normal distribution with mean µ and standard deviation σ is denoted N(µ,σ). (The variance is the square of the standard deviation, σ 2 .) The Standard Normal distribution is when µ=0 and σ=1; its probability density function (pdf) is denoted (x); the cumulative density function (CDF) is Φ(x). This is a graph of the pdf (the height at any point): and this is the CDF:
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One of the basic properties of the normal distribution is that, if X is distributed normally with mean µ and standard deviation σ, then Y = A + bX is also distributed normally, with mean (A + µ) and standard deviation bσ. We will use this particularly when we "standardize" a sample: by subtracting its mean and dividing by its standard deviation, the result should be distributed with mean zero and standard deviation 1. Oppositely, if we are creating random variables with a standard deviation, we can take random numbers with a N(0,1) distribution, multiply by the desired standard deviation, and add the desired mean, to get normal random numbers with any mean or standard deviation. In Excel, you can create normally distributed random numbers by using the RAND() function to generate uniform random numbers on [0,1], then NORMSINV(RAND()) will produce standard-normal-distributed random draws. Wiener Processes Let us propose a very simple model of stock prices: suppose that, at some date, the expected return of a stock is normally distributed with mean zero and some variance. Why mean zero? We have already discussed why the current price of a financial asset should be its expected future value so any additional returns above or below that should have mean zero. We can think of the random part as measuring the amount by which the market's best guess is wrong. So this allows us to use a normal distribution table (or Excel) to figure the probability that the stock will have some value. Since the normal is symmetric, the probability that the stock will be above average is 50%, the same as the probability that the stock will be below average. The probability that it will be more than about 2 standard deviations (well, 1.96 standard deviations)
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