*(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 = m^{2}.

(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 + b^{2})/2]

Variance = exp. [2a + b^{2}]. [exp.[b^{2}] - 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 µ_{2} = µ’_{2} - (Mean)^{2}

**(i) Chi-Square (X ^{2} ) 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, σ^{2} = 2n, (n = d.f.)

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

If X_{i} (i = 1, 2, ... , n) are n independent normal variates with mean µ_{i}; and variance σ^{2}_{i} (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 x^{2} 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, y_{0} 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 X_{i} (i = 1,2, ... , n) be the random samples from a normal population having mean µ and variance σ_{2}, 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) = y_{0}F^{(n/2)-l} ( V_{2} + v1 F)^{-(v1+v2/2)}

where, y_{0} 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)