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Probability Theory in Microsoft Office Build Data Matrix in Microsoft Office Probability Theory

Probability Theory using microsoft toassign data matrix in asp.net web,windows application console app Cov ( X i , X j ) = npi p j (3.68). Figure 3.8 shows a m 2d Data Matrix barcode for None ultinomial distribution with n = 10, p1 = 0.2 and p2 = 0.

3 . Since there are only two free parameters x1 and x2 , the graph is illustrated only using x1 and x2 as axis. Multinomial distributions are typically used with the 2 test that is one of the most widely used goodness-of-fit hypotheses testing procedures described in Section 3.

3.3..

Poisson Distributions Another popular disc Data Matrix 2d barcode for None rete distribution is Poisson distribution. The random variable X has a Poisson distribution with mean ( > 0) if the p.f.

of X has the following form:. e x P( X = x) = f ( x ) = x! 0 Microsoft datamatrix 2d barcode E ( X ) = Var ( X ) = . 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 x 6 7 8 9 10 lambda= 1 lambda= 2 lambda= 4 for x =0,1,2,K otherwise (3.69). The mean and varianc datamatrix 2d barcode for None e of a Poisson distribution are the same and equal : (3.70). Figure 3.9 Three Poisson distributions with = 1, 2, and 4. Figure 3.9 illustrat Microsoft Office data matrix barcodes es three Poisson distributions with = 1, 2, and 4. The Poisson distribution is typically used in queuing theory, where x is the total number of occurrences of some phenomenon during a fixed period of time or within a fixed region of space.

Examples include the number of telephone calls received at a switchboard during a fixed period of. Probability, Statistics, and Information Theory time. In speech recognition, the Poisson distribution is used to model the duration for a phoneme. Gamma Distributions A continuous random Microsoft gs1 datamatrix barcode variable X is said to have a gamma distribution with parameters and ( > 0 and > 0 ) if X has a continuous p.d.f.

of the following form:. 1 x x e f ( x , ) = ( ) 0 . where ( ) = x 1e x dx x >0 x 0 (3.71). (3.72). It can be shown that the function is a factorial function when is a positive integer. (n 1)! ( n) = 1 . 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 0 1 2 3 4 5 x 6 7 8 9 10 alpha = 2 alpha = 3 alpha = 4 n = 2,3,K n=1 (3.73). Figure 3.10 Three Gamma distributions with = 1.0 and = 2.0, 3.0, and 4.0. The mean and varianc e of a gamma distribution are: E( X ) =. and Var ( X ) = 2 . (3.74). Probability Theory Figure 3.10 illustra tes three gamma distributions with = 1.0 and = 2.

0, 3.0, and 4.0.

There is an interesting theorem associated with gamma distributions. If the random variables X 1 ,K , X k are independent and each random variable X i has a gamma distribution with parameters i and , then the sum X 1 + L + X k also has a gamma distribution with parameters 1 + L + k and . A special case of gamma distribution is called exponential distribution.

A continuous random variable X is said to have an exponential distribution with parameters ( > 0 ) if X has a continuous p.d.f.

of the following form:. e x f (x ) = 0 . x>0 x 0 (3.75). It is clear that the Microsoft data matrix barcodes exponential distribution is a gamma distribution with = 1 . The mean and variance of the exponential distribution are: E( X ) = 1 1 and Var ( X ) = 2 . 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 x 6 7 8 9 10 beta = 1 beta =.6 beta =.3 (3.76). Figure 3.11 Three exponential distributions with = 1.0, 0.6 and 0.3. Figure 3.11 shows th ECC200 for None ree exponential distributions with = 1.0, 0.

6, and 0.3. The exponential distribution is often used in queuing theory for the distributions of the duration of a service or the inter-arrival time of customers.

It is also used to approximate the distribution of the life of a mechanical component.. Probability, Statistics, and Information Theory Gaussian Distributions Gaussian distributio n is by far the most important probability distribution mainly because many scientists have observed that the random variables studied in various physical experiments (including speech signals), often have distributions that are approximately Gaussian. The Gaussian distribution is also referred to as normal distribution. A continuous random variable X is said to have a Gaussian distribution with mean and variance 2 ( > 0 ) if X has a continuous p.

d.f. in the following form: f ( x .

, 2 ) = N ( , 2 ) =. ( x )2 exp 2 2 2 (3.77). It can be shown that and 2 are indeed the mean and the variance for the Gaussian distribution. Some examples of Gaussian can be found in Figure 3.4.

The use of Gaussian distributions is justified by the Central Limit Theorem, which states that observable events considered to be a consequence of many unrelated causes with no single cause predominating over the others, tend to follow the Gaussian distribution [6]. It can be shown from Eq. (3.

77) that the Gaussian f ( x . , 2 ) is symmet Microsoft gs1 datamatrix barcode ric with respect to x = . Therefore, is both the mean and the median of the distribution. Moreover, is also the mode of the distribution, i.

e., the p.d.

f. f ( x . , 2 ) attains i ts maximum at the mean point x = . Several Gaussian p.d.

f. s with the same mean , but different variances are illustrated in Figure 3.4.

Readers can see that the curve has a bell shape. The Gaussian p.d.

f. with a small variance has a high peak and is very concentrated around the mean , whereas the Gaussian p.d.

f., with a large variance, is relatively flat and is spread out more widely over the x-axis. If the random variable X is a Gaussian distribution with mean and variance 2 , then any linear function of X also has a Gaussian distribution.

That is, if Y = aX + b , where a and b are constants and a 0 , Y has a Gaussian distribution with mean a + b and variance a 2 2 . Similarly, the sum X 1 + L + X k of independent random variables X 1 ,K , X k , where each random variable X i has a Gaussian distribution, is also a Gaussian distribution..

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