(a) Uniform distribution. Uniform distribution can also be called as continuous distribution or rectangular distribution. The pdf is represented by
f(x)= 1/ (b-a)
a < x < b
Here a and b are two constraints.
Mean = (b+a)/2, variance= (b-a)2/12
This distribution is extensively used in variousmovement flow problems and probabilistic inventory problems. Also (0,1) or (a, b) random numbers performs asignificant role in simulation.
(b) Exponential distribution.This is a continuous distribution, and the pdf is given by
F(x)=1/m. e –x/m
x> 0, m > 0
Here m is the parameter. This is a very advantageous distribution for the theory and will be furthermore discussed in Part B.
(c) Gamma distribution. This is also another continuous distribution and the pdf is represented like,
f(x) = 1/ ba√a.x a-1. e-x/b
x > 0, a > 0, b > 0
Here, √a= gamma
a and b are the variables. Conditional on the values of these variables we acquirenumerousforms of the gamma distribution.
Mean = ab and Variance = ab2
When a = 1, we catch exponential distribution.
(d) Beta distribution– This is also another continuous distribution and the pdf is prearranged by
, 0 <x< 1, a> 0, b > 0
= 0, otherwise
Similarly herea and b are two variables of the concerned distribution. When a = 1 and b = 1, we achieve the uniform distribution in the interval (0, 1).
Mean = a/ (a+b),
Variance= ab/ (a+b)2(a+b+1)
(e) Geometric distribution. One of the discrete distribution type. The chance of receiving the first attainment on the (x + 1 )th trial is expressed by
P(x) = pcf, for x = 0, 1, 2,…
Where, p = Probability of success in any trial
q = 1- p
Mean= q/p and variance= q/p2
(j) Log-Normal distribution. If the logarithm of any random variable representsageneral distribution, then the distribution assumed by the random variable can be known as Log-Normal distribution. Its probability density function will be expressed as:
f(x)= 1/b√2π.x-1.exp [-(ln x-a)2/ 2b2], x>0, b>0
= 0, otherwise
Mean = exp. [(a + b2)/2]
Variance = exp. [2a + b2]. [exp.[b2] – 1]
(g) Hypergeometric distribution. Let us assume a sample of n units to be taken out from a package of containing N units, of which d are defective.
Therefore the x attainments of the defective samples and n – x failures can be selected in (d x) (N-dn-x) ways.
Also, n items can be chosen from a set of N objects in (N n) ways.
For packaging without replacement is the probability of getting “x successes in n trials” is
p (x, n, d, N) =(d x) (N-dn-x) / (N n) for x = 0, 1, … , n
Mean= n. d/N
This distribution approaches to binomial with p = d/N when N ->∞ .
(h) Weibull distribution. This is the other mostimperative distribution that is used in consistency and life testing of an item. The probability density function is represented by
Where, α and β are two variables of the distribution.
Using integration by parts,
So the for this expression will be,
Variance µ2 = µ’2 – (Mean)2
(i) Chi-Square (X2 ) distribution- weconsider a variable X which follows the chi-square distribution if its pdf is of the form
F(x)= 1/(n/2).e-x/2.x(n/2)-1, 0<x<∞
The variable n is a positive integer, and that is called the number of degrees of freedom.
A variable that will follow the chi-square distribution correctly will be called as a chi-square variate.
- The chi-square curve has to be positively skew.
- Mean = n, σ2 = 2n, (n = d.f.)
- If x2 is a chi-square variate with ndf., 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 x2=Σi (Xi-µi/σi)is a chi-square variate with ndf
- M.G.F. = (1 – 2t)-n/2, |2t| < 1
- In practice, for n ≥ 30, the x2 distribution is estimated by normal distribution.
(j) Student’s t-distribution. A random variable iffollows the student’s t-distribution or just -distribution if its pdf is given by
f(t) = y0 (1+ t2/n)-(n+1)/2 ,- ∞ < t <∞
Where, y0 is a constant variable so that the area under the curve is unity and n are called degrees of freedom.
- The t-curve is symmetrical about 0, and leptokurtic i.e., β1 = 0,
- Mean = 0, Variance =n/(n-2), (n > 2)
- For large df, the t-distribution can be approximated by the standard normal distribution.
- 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
n is the degrees of freedom.
(i) The F-curve is positively skew.
(ii) Mean = v2/ (v2 – 2)
Variance = 2(v1 + v2 – 2)/v1(v2-4). (v2/v2-2)2
(iii) If y1 and y2 are independent chi-square variates with n1 and n2 degrees of freedom, then we are left with= y1/n1/ y2/n2, follows F-distribution with (n1, n2) df.
Links of Previous Main Topic:-
- Introduction to statistics
- Knowledge of central tendency or location
- Definition of dispersion
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution
Links of Next Statistics Topics:-
- What is sampling
- Statistical hypothesis and related terms
- Analysis of variance introduction
- Definition of stochastic process
- Introduction operations research
- Introduction and mathematical formulation in transportation problems
- Introduction and mathematical formulation
- Queuing theory introduction
- Inventory control introduction
- Simulation introduction
- Time calculations in network
- Introduction of game theory