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Other Distributions

(a) Uniform distribution. This is a continuous distribution and is also known as rectangular distribution. The pdf  is given by

a < x < b

= 0,                        otherwise

Here a and b are two parameters.

 

This distribution is widely used in many traffic flow problems and probabilistic inventory problems. Also (0,1) or (a, b) random numbers play an important role in simulation.

 

(b) Exponential distribution. This is a continuous distribution and the pdf is given by

 

x > 0, m > 0

otherwise

Here m is the parameter. A very useful distribution for queuing theory and again will be discussed in Part B.

 

Mean = m, Variance = m2.

 

(c) Gamma distribution. This is a continuous distribution and the pdf is given by

 

x > 0, a > 0, b > 0

                                                          

                 =0                                 otherwise

  = gamma (a) and a, b are two parameters. Depending on the values of these parameters we obtain various shapes of the gamma distribution.

 

Mean = ab    and          Variance = ab2

 

when a = 1, we get exponential distribution.

 

(d) Beta distribution. This is a continuous distribution and the pdf is given by

 

,     0 <x< 1, a> 0, b > 0

 

            = 0,   otherwise

where, a and b are two parameters of this distribution. When a = 1 and b = 1, we obtain the uniform distribution in the interval (0, 1).

 

 

                                                Mean = a/ (a+b),

 

(e) Geometric distribution. This is a discrete distribution. The probability of getting the first success on the (x + 1 )th trial is given by

 

P(x) = pcf, for x = 0,1,2, ...

 

where,                         p = Probability of success in any trial

 

q =  1- p

 

(j) Log-Normal distribution. If the logarithm of a random variable shows normal distribution, then the distribution given by the random variable is called Log-Normal distribution. Its probability density function is given by

Mean = exp. [(a + b2)/2]

Variance = exp. [2a + b2]. [exp.[b2] - 1]

 

(g) Hypergeometric distribution. Consider a sample of n units to be drawn from a lot containing N units, of which d are defective.

The x successes (defectives) and n - x failures can be chosen in ways.

Also n items can be chosen from a set of N objects in ways. For sampling without replacement

 

the probability of getting "x successes in n trials" is

p (x, n, d, N) =       for x = 0, 1, ... , n

 

 

 This distribution approaches to binomial with p = d/N when N -> ∞ .

 

(h) Weibull distribution. This is an important distribution used in reliability and life testing of an item. The probability density function is given by

where, α and β are two parameters of the distribution.

Using integration by parts,

 

Therefore,

Variance µ2 = µ’2 - (Mean)2

 

 

(i) Chi-Square (X2 ) distribution. A random variable X is said to follow chi-square distribution if its pdf is of the form

 

 

The parameter n (positive integer) is called the number of degrees of freedom.

A variable which follows chi-square distribution is called a chi-square variate.

 

Properties:

(i)                 The chi-square curve is positively skew.

(ii)               Mean = n, σ2 = 2n, (n = d.f.)

(iii)             If x2 is a chi-square variate with n d.f., then x2/2 is a Gamma variate with parameter (n/2).

If Xi (i = 1, 2, ... , n) are n independent normal variates with mean  µi; and variance σ2i (i = 1,2, ... ,n) then

is a chi-square variate with n df

 

 

 

            (v) M.G.F. = (1 - 2t)-n/2,            |2t| < 1

 

(vi)  In practice, for n ≥ 30, the x2 distribution is approximated by normal distribution.

(j) Student's t-distribution. A random variable is said to follow student's t-distribution or simply -distribution if its pdf is given by

 

 

,- ∞ < t <∞

where, y0 is a constant such that the area under the curve is unity and n is called degrees of freedom.

 

Properties :

(i)                 The t-curve is symmetrical about 0, and leptokurtic i.e., β1 = 0, β2 > 3.

(ii)               Mean = 0, Variance =n/(n-2),           (n > 2)

(iii)             For large df, the t-distribution can be approximated by the standard normal distribution.

(iv)             Let Xi (i = 1,2, ... , n) be the random samples from a normal population having mean µ and variance σ2, then the statistic

(k) F-distribution; A random variable is said to be F-distribution with degrees of freedom (v1, v2) if its pdf is of the form

 

f (F) = y0F(n/2)-l ( V2 + v1 F)-(v1+v2/2)

 

where, y0 is a constant such that the area under the curve is unity.

 

Properties: 

(i) The F-curve is positively skew.

 

               (ii)     Mean = v2/ (v2 – 2)