**(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

= 0,

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 ^{â€“x/m}

=0

x> 0, m > 0

Otherwise

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/ b^{a}âˆša.x ^{a-1}. e^{-x/b}

x > 0, a > 0, b > 0

=0Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â otherwise

Here, âˆša= gamma

a and b are the variables. Conditional on the values of these variables we acquirenumerousforms of the gamma distribution.

Mean = ab Â Â Â andÂ Â Â Â Â Â Â Â Â Variance = ab2

When a = 1, we catch exponential distribution.

**(d) Beta distribution**– This is also another continuous distribution and the *pdf* is prearranged by

f(x)=âˆša+b/âˆša.âˆšb. x^{a-1}(1-x)^{b-1}

,Â Â Â Â 0 <x< 1, a> 0, b > 0

= 0,Â Â 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,Â p = Probability of success in any trial

q = 1- p

Mean= q/p and variance= q/p^{2}

**(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/bâˆš2Ï€.x^{-1}.exp [-(ln x-a)^{2}/ 2b^{2}], x>0, b>0

= 0,Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â otherwise

Mean = exp. [(a + b^{2})/2]

Variance = exp. [2a + b^{2}]. [exp.[b^{2}] – 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-d}_{n-x})Â Â 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-d}_{n-x})Â / (^{N }_{n}) Â Â Â Â Â for x = 0, 1, … , n

Mean= n. d/N

Variance= n.d.(N-d).(N-n)/N^{2}.(N-1)

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

**(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,Â Î± and Î² are two variables of the distribution.

Using integration by parts,

So the for this expression will be,

Variance Âµ_{2} = Âµâ€™_{2} – (Mean)^{2}

**(i) Chi-Square (X ^{2} ) 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},Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â 0<x<âˆž

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.

**Properties:**

- The chi-square curve has to be positively skew.
- Mean = n, Ïƒ
^{2}= 2n, (n = d.f.)

- If x2 is a chi-square variate with ndf., 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 meanÂ Âµ_{i}; and variance Ïƒ^{2}_{i}(i = 1,2, … ,n) then x2=Î£_{i}(X_{i}-Âµ_{i}/Ïƒ_{i})is a chi-square variate with ndf - M.G.F. = (1 – 2t)
^{-n/2},Â Â Â Â Â Â Â Â Â Â Â |2t| < 1 - In practice, for n â‰¥ 30, the x
^{2}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) = y_{0} (1+ t2/n)^{-(n+1)/2Â Â Â Â Â },- âˆž < t <âˆž

Where, y_{0} is a constant variable so that the area under the curve is unity and n are called degrees of freedom.

**Properties:**

- The t-curve is symmetrical about 0, and leptokurtic i.e., Î²1 = 0,
- Mean = 0, Variance =n/(n-2),Â Â Â Â Â Â Â Â Â Â (n > 2)
- For large df, the t-distribution can be approximated by the standard normal distribution.
- Let X
_{i}(i = 1,2, …, n) be the random samples from a normal population having mean Âµ and variance Ïƒ_{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

n is the degrees of freedom.

**Properties:Â **

(i) The F-curve is positively skew.

(ii) Mean = v2/ (v2 â€“ 2)

Variance = 2(v_{1} + v_{2} â€“ 2)/v_{1}(v_{2}-4). (v_{2}/v_{2}-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
- Moments
- Bivariate distribution
- Theorem of total probability addition theorem
- Random variable
- Binomial distribution

**Links of NextÂ Statistics Topics:-**

- What is sampling
- Estimation
- 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