NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS in .NET

Display QR Code 2d barcode in .NET NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS
4-7 NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS
Visual .net qr code 2d barcode encoding with .net
use visual .net qr bidimensional barcode creator toinsert qr-codes with .net
We began our section on the normal distribution with the central limit theorem and the normal distribution as an approximation to a random variable with a large number of trials. Consequently, it should not be a surprise to learn that the normal distribution can be used to approximate binomial probabilities for cases in which n is large. The following example illustrates that for many physical systems the binomial model is appropriate with an extremely large value for n. In these cases, it is dif cult to calculate probabilities by using the binomial distribution. Fortunately, the normal approximation is most effective in these cases. An illustration is provided in Fig. 4-19. The area of each bar equals the binomial probability of x. Notice that the area of bars can be approximated by areas under the normal density function.
Visual Studio .NET qr-codes reader on .net
Using Barcode scanner for .NET Control to read, scan read, scan image in .NET applications.
4-7 NORMAL APPROXIMATION TO THE BINOMIAL AND POISSON DISTRIBUTIONS
decoding barcode for .net
Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in Visual Studio .NET applications.
0.25 n = 10 p = 0.5 0.20
.NET Crystal bar code encoder with .net
using barcode maker for .net framework crystal control to generate, create barcode image in .net framework crystal applications.
0.15 f(x) 0.10 0.05
Denso QR Bar Code barcode library for .net c#
generate, create qrcode none for visual c# projects
Figure 4-19 Normal approximation to the binomial distribution.
Control denso qr bar code image with .net
using web form toreceive denso qr bar code in asp.net web,windows application
0.00 0 1 2 3 4 5 x 6 7 8 9 10
Control quick response code image in visual basic.net
using barcode generation for vs .net control to generate, create qr codes image in vs .net applications.
EXAMPLE 4-17
.net Framework barcode generator in .net
generate, create bar code none with .net projects
In a digital communication channel, assume that the number of bits received in error can be modeled by a binomial random variable, and assume that the probability that a bit is received in error is 1 10 5. If 16 million bits are transmitted, what is the probability that more than 150 errors occur Let the random variable X denote the number of errors. Then X is a binomial random variable and P 1X 1502 1 P1x 1502 1 aa
Bar Code integrating for .net
use visual .net crystal bar code implementation toadd bar code for .net
0 150
Assign matrix barcode with .net
using .net vs 2010 tocreate 2d barcode for asp.net web,windows application
16,000,000 b 110 5 2 x 11 x
VS .NET Crystal pdf417 drawer with .net
using barcode encoding for visual .net crystal control to generate, create pdf 417 image in visual .net crystal applications.
10 5 2 16,000,000
USPS PLANET Barcode generation on .net
using .net vs 2010 crystal toadd planet with asp.net web,windows application
Clearly, the probability in Example 4-17 is dif cult to compute. Fortunately, the normal distribution can be used to provide an excellent approximation in this example.
Insert data matrix ecc200 with visual basic
using an asp.net form crystal toadd datamatrix for asp.net web,windows application
Normal Approximation to the Binomial Distribution
2d Matrix Barcode barcode library for .net
generate, create 2d matrix barcode none in .net projects
If X is a binomial random variable, Z
UPC-A Supplement 5 barcode library for c#
generate, create upc-a none with c# projects
X np 1np11 p2
Control ean-13 data on office excel
to insert european article number 13 and ean-13 supplement 5 data, size, image with office excel barcode sdk
(4-12)
Control upc symbol data with microsoft word
to print upca and upc a data, size, image with word documents barcode sdk
is approximately a standard normal random variable. The approximation is good for np 5 and n11 p2 5
Asp.net Aspx Crystal qr maker in vb
using asp.net web service crystal todraw qr-code for asp.net web,windows application
Recall that for a binomial variable X, E(X) np and V(X) np(1 p). Consequently, the expression in Equation 4-12 is nothing more than the formula for standardizing the random variable X. Probabilities involving X can be approximated by using a standard normal distribution. The approximation is good when n is large relative to p.
Control code 128b data on microsoft word
code 128 code set a data on word documents
CHAPTER 4 CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS
Control ean / ucc - 13 image for microsoft word
generate, create ean / ucc - 13 none on microsoft word projects
EXAMPLE 4-18
The digital communication problem in the previous example is solved as follows: P1X 1502 Pa P1Z Because np 116 106 211 10 5 2 is expected to work well in this case. X 160 10 2
216011 150
160 0.785
10 5 2
160 and n(1
p) is much larger, the approximation
EXAMPLE 4-19
Again consider the transmission of bits in Example 4-18. To judge how well the normal approximation works, assume only n 50 bits are to be transmitted and that the probability of an error is p 0.1. The exact probability that 2 or less errors occur is P1X 22 a 50 b 0.950 0 50 a b 0.110.949 2 1 50 a 2 b 0.12 10.948 2 0.112
Based on the normal approximation P1X 22 Pa X 5 2.12 2 5 b 2.12 P1Z 1.422 0.08
Even for a sample as small as 50 bits, the normal approximation is reasonable. If np or n(1 p) is small, the binomial distribution is quite skewed and the symmetric normal distribution is not a good approximation. Two cases are illustrated in Fig. 4-20. However, a correction factor can be used that will further improve the approximation. This factor is called a continuity correction and it is discussed in Section 4-8 on the CD.
0.4 n p 10 0.1 10 0.9 0.3
f(x)
Figure 4-20 Binomial distribution is not symmetrical if p is near 0 or 1.
0.0 0 1 2 3 4 5 x 6 7 8 9 10
4-7 NORMAL APPROXIMATION TO THE BIOMIAL AND POISSON DISTRIBUTIONS
hypergometric distribution
Figure 4-21
binomial distribution
np 5 n11 p2 5
normal distribution
Conditions for approximating hypergeometric and binomial probabilities.
Recall that the binomial distribution is a satisfactory approximation to the hypergeometric distribution when n, the sample size, is small relative to N, the size of the population from which the sample is selected. A rule of thumb is that the binomial approximation is effective if n N 0.1. Recall that for a hypergeometric distribution p is de ned as p K N. That is, p is interpreted as the number of successes in the population. Therefore, the normal distribution can provide an effective approximation of hypergeometric probabilities when n N 0.1, np 5 and n(1 p) 5. Figure 4-21 provides a summary of these guidelines. Recall that the Poisson distribution was developed as the limit of a binomial distribution as the number of trials increased to in nity. Consequently, it should not be surprising to nd that the normal distribution can also be used to approximate probabilities of a Poisson random variable.