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50 while the Recognizing QR Code In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. QR Maker In None Using Barcode creation for Software Control to generate, create QR Code 2d barcode image in Software applications. The Law of Large Numbers for Bernoulli Trials
Scanning Denso QR Bar Code In None Using Barcode recognizer for Software Control to read, scan read, scan image in Software applications. Denso QR Bar Code Generation In C#.NET Using Barcode encoder for Visual Studio .NET Control to generate, create Quick Response Code image in .NET applications. 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 41 (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) QR Code JIS X 0510 Maker In .NET Using Barcode creation for ASP.NET Control to generate, create Denso QR Bar Code image in ASP.NET applications. QR Code Printer In VS .NET Using Barcode creator for Visual Studio .NET Control to generate, create QR Code image in .NET framework applications. 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. Paint QRCode In VB.NET Using Barcode printer for Visual Studio .NET Control to generate, create QR Code image in Visual Studio .NET applications. Printing Code128 In None Using Barcode creation for Software Control to generate, create Code 128 image in Software applications. The Normal Distribution
GTIN  13 Maker In None Using Barcode generator for Software Control to generate, create GS1  13 image in Software applications. Barcode Drawer In None Using Barcode drawer for Software Control to generate, create bar code image in Software applications. 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) USS128 Printer In None Using Barcode generator for Software Control to generate, create EAN / UCC  13 image in Software applications. Data Matrix 2d Barcode Generation In None Using Barcode printer for Software Control to generate, create Data Matrix ECC200 image in Software applications. x 1 3 `e s!2p (v m)2/2s2
Create OneCode In None Using Barcode drawer for Software Control to generate, create USPS OneCode Solution Barcode image in Software applications. Drawing UPCA Supplement 5 In None Using Barcode printer for Microsoft Word Control to generate, create UPC Symbol image in Office Word applications. where and given by
Painting Barcode In None Using Barcode drawer for Font Control to generate, create bar code image in Font applications. EAN13 Drawer In .NET Framework Using Barcode creation for Visual Studio .NET Control to generate, create GS1  13 image in .NET framework applications. 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) Painting Barcode In None Using Barcode creator for Office Excel Control to generate, create bar code image in Office Excel applications. ANSI/AIM Code 39 Creator In .NET Using Barcode drawer for Reporting Service Control to generate, create Code 3/9 image in Reporting Service applications. CHAPTER 4 Special Probability Distributions
Painting ANSI/AIM Code 39 In None Using Barcode generator for Online Control to generate, create USS Code 39 image in Online applications. Code 39 Extended Drawer In .NET Framework Using Barcode creation for Visual Studio .NET Control to generate, create USS Code 39 image in .NET applications. 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. 41. 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. 41 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 42 we list some important properties of the general normal distribution.
Table 42 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

