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29.13 PARETO DISTRIBUTION

The Pareto CDF is a power curve that can be easily fit to observed data. The key characteristics of Pareto distribution are summarized in Table 29.13.

Pareto distribution is useful to fit a distribution to observed data. Given a sample of n observations {x1, x2,..., xn}, the maximum likelihood estimate of the parameter a is

The inverse transformation method is the easiest way to generate Pareto variates. Generate u ~ U(0, 1) and return 1/u1/a.

29.14 PASCAL DISTRIBUTION

The Pascal distribution is an extension of the geometric distribution. In a sequence of Bernoulli trials, the number of trials up to and including the mth success has a Pascal distribution. The key characteristics of the Pascal distribution are summarized in Table 29.14.

The Pascal distribution is used to model the number of attempts to get a certain number of successes; for example:

1.  Number of attempts to transmit an m-packet message
2.  Number of bits to be sent to successfully receive an m-bit signal

TABLE 29.14 Pascal Distribution Pascal (p,m)


1.  Parameters: p = probability of success, 0 > p > 1
m = number of successes; m should be a positive integer
2.  Range: x = m, m + 1,...,∞
3.  pmf:
4.  Mean: m/p
5.  Variance: m(1 – p)p/p2

The attempts must be independent and identical as explained under “Bernoulli Distribution” in Section 29.1.

To generate Pascal variates, generate m geometric variates G(p) and return their sum as Pascal(p,m).

29.15 POISSON DISTRIBUTION

The Poisson distribution is a limiting form of the binomial distribution, and it is used extensively in queueing models. The key characteristics of Poisson distribution are summarized in Table 29.15.

TABLE 29.15 Poisson Distribution Poisson(λ)


1.  Parameters: λ = mean, λ > 0
2.  Range: x = 0,1,2,...,∞
3.  pmf:
4.  Mean: λ
5.  Variance: λ

The Poisson distribution is used to model the number of arrivals over a given interval; for example:

1.  Number of requests to a server in a given time interval t
2.  Number of component failures per unit time
3.  Number of queries to a database system over t seconds
4.  Number of typing errors per form

The Poisson distribution is particularly appropriate if the arrivals are from a large number of independent sources. Such arrival processes are also called Poisson processes and are discussed further in Chapter 30 on queueing theory.

See Section 29.3 for choosing among binomial, negative binomial, and Poisson distributions based on the relative magnitude of mean and variance.

Poisson variates can be generated as follows:

1.  Inverse Transformation Method: Compute the CDF F(x) for x = 0, 1, 2,... up to a suitable cutoff and store in an array. For each Poisson random variate, generate a U(0,1) variate u, and search the array to find x such that F(x) ≤ u < F(x + 1); return x.
2.  Starting with n = 0, generate un ~ U(0,1) and compute the product . As soon as the product becomes less than e λ, return n as the Poisson(λ) variate. Note that n is such that u0u1 ... un–1 > e λ ≥ u0u1 ... un. On the average, λ + 1 uniform variates are required per Poisson variate.

29.16 STUDENT’S t DISTRIBUTION

This distribution was derived by W. S. Gosset (1876–1937), a statistician for a winery whose owner did not appreciate his publishing. Gosset, therefore, published his paper under the pseudonym Student. The symbol t was used to denote the variable and hence the name “Student’s t distribution.” The key characteristics of the t distribution are summarized in Table 29.16. If x ~ N(0, 1) is a unit normal variate and y ~ X2(v) is a chi-square variate, the ratio has a t distribution with v degrees of freedom:

The pdf of a t-variate is very similar to that of a unit normal. The distribution is bell shaped and is symmetrical about zero. For large degrees of freedom (v > 30), a t distribution can be approximated by a unit normal.

The t distribution is used whenever a ratio of a normal variate and the square root of a chi-square variable is involved and is commonly used in setting confidence intervals and in t-tests as discussed in Chapter 13.

The t-variates can be generated using characterization as follows. Generate x ~ N(0,1) and y ~ X2(v) and return

TAKE 29.16 Student’s t-Distribution t(v)


1.  Parameters: v = degrees of freedom; v must be a positive integer
2.  Range: – ∞ ≤ x ∞ ≤
3.  pmf:
4.  Variance: v/(v – 2), for v > 2.

29.17 UNIFORM DISTRIBUTION (CONTINUOUS)

This is one of the simplest distributions to use. The key characteristics of the uniform distribution are summarized in Table 29.17.

A uniform distribution is commonly used if a random variable is bounded and no further information is available; for example:

1.  Distance between source and destinations of messages on a network
2.  Seek time on a disk

TABLE 29.17 Uniform Distribution (Continuous) U (a,b)


1.  Parameters: a = lower limit
b = upper limit b > a
2.  Range axb
3.  pdf:
4.  CDF:
5.  Mean:
6.  Variance: (b - i)2/12

To generate U(a,b), generate u ~ U(0, 1) and return a + (ba)u.

29.18 UNIFORM DISTRIBUTION (DISCRETE)

This is a discrete version of the uniform distribution. It takes a finite number of values, each with the same probability. The key characteristics of this distribution are summarized in Table 29.18.

The discrete uniform distribution is used when it is believed that the value is equally likely over a bounded interval; for example:

1.  Track numbers for seeks on a disk
2.  The I/O device number selected for the next I/O
3.  The source and destination node for the next packet on a network

To generate UD(m,n), generate u ~ U(0, 1), return [m + (nm + 1)u].

TABLE 29.18 Uniform Distribution (Discrete) UD(m,n)


1.  Parameters: m = lower limit; m must be an integer
n = upper limit n must be an integer greater than m
2.  Range x = m,m + 1,m + 2,...,n
3.  pmf:
4.  CDF:
5.  Mean: (n + m)/2
6.  Variance:

29.19 WEIBULL DISTRIBUTION

The Weibull distribution is commonly used in reliability analysis. The key characteristics of a Weibull distribution are summarized in Table 29.19. If b = 3.602, the Weibull distribution is close to a normal. For b > 3.602, it has a long left tail. For b < 3.602, it has a long right tail. For b ≤ 1, the Weibull pdf is L shaped, and for b > 1, it is bell shaped. For large b, the Weibull pdf has a sharp peak at the mode.

The Weibull distribution is used to model lifetimes of components. For b < 1, the Weibull distribution gives a failure rate increasing with time. For b > 1, the failure rate decreases with time. At b = 1, the failure rate is constant and the lifetimes are exponentially distributed.

The inverse transformation technique can be used to generate Weibull variates. Generate u ~ U(0, 1) and return a(ln u)1/b as Weibull(a,b).

TABLE 29.19 Weibull Distribution Weibull (a,b)


1.  Parameters: a = scale parameter a > 0
b = shape parameter b > 0
2.  Range: 0 ≤ x ≤ ∞
3.  pdf:
4.  CDF: F(x) = 1 - e-(x/a)b
5.  Mean:
6.  Variance:


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