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The Law of Large Numbers for Bernoulli Trials
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The law of large numbers, page 83, has an interesting interpretation in the case of Bernoulli trials and is presented in the following theorem. Theorem 4-1 (Law of Large Numbers for Bernoulli Trials): Let X be the random variable giving the number of successes in n Bernoulli trials, so that X>n is the proportion of successes. Then if p is the probability of success and is any positive number, X lim Pa 2 n p2 Pb 0 (3)
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In other words, in the long run it becomes extremely likely that the proportion of successes, X>n, will be as close as you like to the probability of success in a single trial, p. This law in a sense justifies use of the empirical definition of probability on page 5. A stronger result is provided by the strong law of large numbers (page 83), which states that with probability one, lim X>n p, i.e., X>n actually converges to p except in a negligible nS` number of cases.
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The Normal Distribution
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One of the most important examples of a continuous probability distribution is the normal distribution, sometimes called the Gaussian distribution. The density function for this distribution is given by 1 (4) e (x m)2/2s2 ` x ` s22p are the mean and standard deviation, respectively. The corresponding distribution function is f (x) F(x) P(X x)
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x 1 3 `e s!2p (v m)2/2s2
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where and given by
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If X has the distribution function given by (5), we say that the random variable X is normally distributed with mean and variance 2. If we let Z be the standardized variable corresponding to X, i.e., if we let Z X s m (6)
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CHAPTER 4 Special Probability Distributions
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then the mean or expected value of Z is 0 and the variance is 1. In such cases the density function for Z can be 0 and 1, yielding obtained from (4) by formally placing 1 (7) e z2>2 22p This is often referred to as the standard normal density function. The corresponding distribution function is given by f (z)
z z 1 1 1 u2>2 du u2>2 du (8) 3 `e 30 e 2 !2p !2p We sometimes call the value z of the standardized variable Z the standard score. The function F(z) is related to the extensively tabulated error function, erf(z). We have
F(z)
erf(z)
2 z e !p 30
F(z)
1 c1 2
erf a
z bd !2
A graph of the density function (7), sometimes called the standard normal curve, is shown in Fig. 4-1. In this graph we have indicated the areas within 1, 2, and 3 standard deviations of the mean (i.e., between z 1 and 1, z 2 and 2, z 3 and 3) as equal, respectively, to 68.27%, 95.45% and 99.73% of the total area, which is one. This means that P( 1 Z 1) 0.6827, P( 2 Z 2) 0.9545, P( 3 Z 3) 0.9973 (10)
Fig. 4-1
A table giving the areas under this curve bounded by the ordinates at z 0 and any positive value of z is given in Appendix C. From this table the areas between any two ordinates can be found by using the symmetry of the curve about z 0.
Some Properties of the Normal Distribution
In Table 4-2 we list some important properties of the general normal distribution.
Table 4-2
Mean Variance Standard deviation Coefficient of skewness Coefficient of kurtosis Moment generating function Characteristic function M(t) f(v)
3 4 2
0 3 eut eimv
(s2t2>2) (s2v2>2)
CHAPTER 4 Special Probability Distributions
Relation Between Binomial and Normal Distributions
If n is large and if neither p nor q is too close to zero, the binomial distribution can be closely approximated by a normal distribution with standardized random variable given by Z X np !npq (11)
Here X is the random variable giving the number of successes in n Bernoulli trials and p is the probability of success. The approximation becomes better with increasing n and is exact in the limiting case. (See Problem 4.17.) In practice, the approximation is very good if both np and nq are greater than 5. The fact that the binomial distribution approaches the normal distribution can be described by writing lim P aa S