Statistical Methods And Business Economics For Operations Research

Discrete Probability Distributions

Discuss about the Statistical Methods and Business Economics for Operations Research.

There are different types of distributions in statistics. The distributions can mainly be classified into two categories; discrete distribution and continuous distribution. These two categories contain various types of distributions that are used in different statistical analysis.

These distributions are to be considered in this assignment and they are to be explained in statistical ways. These distributions are useful in various statistical calculations. The distributions are mainly probability distributions. These are regarded as discrete probability distribution and continuous probability distribution. Information regarding these two distributions and the distributions that fall under these two categories would be explored in this assignment.

There are two types of distributions; discrete distribution and continuous distribution. Considering discrete distributions, various types of discrete distributions would be studied under this topic:

Binomial distribution- binomial distribution is a discrete distribution that measures the number of success over a given number of trails having a specific success probability. This distribution has two parameters; n and p[1]. N is the number of trails and p is the probability of success. The probability mass function of binomial distribution is given by (ⁿC_k) p^k (1-p)^(n-k) ; where k is the number of success.

Poisson distribution- A discrete random variable X is said to follow a Poisson distribution with parameter λ > 0, if the probability mass function is given by λ^x e^{– λ} / x!. Poisson distribution is considered when the X is a discrete event occurring independently and the rate of occurring of the event is constant[2]. This distribution is also considered when the probability of an event in any interval is proportion to the length of the interval.

Hypergeometric distribution- This discrete probability distribution describes the probability of k success in n trails and each daw is either a success or a failure. The samples are drawn without replacement from a population of size N that contains exactly K success. The probability mass function of this distribution is ( ^KC_k) (^N-K C _n-k) / (^NC_n ).

Bernoulli distribution- A random variable is said to follow Bernoulli distribution if the random variable takes the value 1 with probability p and 0 with probability q = 1-p[3]. This type of probability distribution is used when there are only two outcome of the trial. This distribution is considered as a special case of two point distribution.

Borel distribution- This discrete probability distribution arise due to branching process and queuing theory. When the number of offspring of any organism is said to follow Poisson distribution and the mean of the offspring produced is not more than 1, there is a chance of extinction of the descendents of each of the individuals. The random variable is said to follow Borel distribution if the random variable represents the number of descendents of that individual in the above situation. The probability mass function of this distribution is given as e^–μn (μn)^(n-1) / n!.

Binomial distribution

Geometric distribution- Geometric distribution is considered as either of the two discrete probability distributions. Firstly, supporting on the set {1, 2, 3,……}, the probability distribution of X following Bernoulli distribution must get one success. Secondly, if the random variable represents the failures before the first success, the probability distribution is said to follow geometric distribution. The probability mass function is given by (1 – p)^{(k – 1)} p; where p represents the probability distribution[4].

Considering continuous distributions, various types of continuous distributions would be studied under this topic. The following continuous probability distributions are explored in this assignment.

Normal distribution- Normal distribution is the most common continuous distribution, which is commonly used in natural science and social science. Due to central limit theorem, the normal distribution is the most commonly used continuous distribution in statistics[5]. According to central limit theorem, when the random variables are drawn independently and the number of variables are sufficiently large, the distribution of the independent variables converge to normal distribution. The probability distribution function of normal distribution is a given by (1/sqrt(2σ² π)) e ^{– (x-μ)*( x-μ)/ 2σ*σ} . The curve of normal distribution is bell shaped; the mean of the distribution id given by μ and the variance is given by σ².

Uniform distribution- It is also known as rectangular distribution as it have a constant probability distribution. Continuous distribution is a family of symmetric probability distribution where each member of the family has intervals of same length and is equal probable. The support of this distribution is defined by two variables, “a” and “b” which defines the minimum and maximum values respectively. The probability distribution function of uniform distribution is given by 1/ (a-b) for x e[a, b].

Chi- squared distribution- When k independent standard normal variables are considered, the distribution of the sum if squares of these k independent variables follow chi squared distribution. Chi squared distribution is used in case of goodness of fit of an observed variable, to test the independence of two criteria for qualitative data. Suppose X₁, X₂, ….., X_k are independent standard normal random variables, then the sum of the squares of these variables Q = Σ X²_i is distributed as chi squared distribution with k degrees of freedom[6]. The probability distribution function of chi squared theory is given by (1/ (2^k/2 Γ (k/2)) * x^(k/2)-1 e ^{– x/2}.

Student’s T-distribution- A variable is said to follow Student’s T-distribution if it is formed by the ratio of a standard normally distributed random variable and the square root of a chi-square variable distributed independently and is divided by its degree of freedom. This denotes the random variable of student’s t-distribution is given by X = Z / (sqrt(U/k)); where Z is distributed as standard normal distribution, U is distributed as chi-square distribution and k is the degrees of freedom of chi-square variable[7]. The probability mass function of the variable is given by Γ[(k+1) / 2]/ Γ(k / 2) * 1/ sqrt(k*Π) * 1/ (1 + x²/k) ^(k+1)/2.

Poisson distribution

Exponential distribution – A random variable X is said to follow exponential distribution if its density is given by λe^–λx[8]. Exponential is another continuous distribution that has been used as a model for lifetimes of various things.

Gamma distribution- When a random variable X has the density function as λ / Γ(r) * (λx) ^(r-1) e^–λx; the variable X is said to follow gamma distribution. When r > 0 and λ > 0[9]. Exponential distribution is a special case of gamma distribution. Gamma distribution is mainly used in Bayesian statistics where the distribution of the prior distribution is found to be gamma distribution.

Cauchy distribution- Cauchy distribution is a commonly used continuous probability distribution as the canonical example of a “pathology distribution”. This is because both the the mean and the variance of this distribution is undefined. Cauchy distribution does not have any finite moments and thus it has no moment generating functions[10]. The shape of the Cauchy distribution can be defined as distribution of the horizontal intercept of any ray that is generating from (x₀, ϒ) and the distribution of the angle is uniform. The probability mass function of Cauchy distribution is given by (1/ Πϒ) * [ ϒ² / ((x-x₀)² + ϒ²)].

Conclusion

It was seen in this assignment that there are two types of distributions; discrete probability distribution and continuous probability distributions. There are various distributions under these two types of distributions. Every distribution has own probability mass function and probability density function.  They are used in various statistical analyses and the uses of the distributions are given in the assignment.

References

Ben-Israel, A., 2013. A concentrated Cauchy distribution with finite moments. Annals of Operations Research, pp.1-7.

Bueno-Delgado, M.V., Ferrero, R., Gandino, F., Pavon-Marino, P. and Rebaudengo, M., 2013. A geometric distribution reader anti-collision protocol for RFID dense reader environments. IEEE Transactions on Automation Science and Engineering, 10(2), pp.296-306.

Collins, D.J., Neild, A., Liu, A.Q. and Ai, Y., 2015. The Poisson distribution and beyond: methods for microfluidic droplet production and single cell encapsulation. Lab on a Chip, 15(17), pp.3439-3459.

Dai, B., Ding, S. and Wahba, G., 2013. Multivariate bernoulli distribution.Bernoulli, 19(4), pp.1465-1483.

Hong, Y., 2013. On computing the distribution function for the Poisson binomial distribution. Computational Statistics & Data Analysis, 59, pp.41-51.

Krishnamoorthy, K., Mathew, T. and Mukherjee, S., 2012. Normal-based methods for a gamma distribution. Technometrics.

Ristić, M.M. and Balakrishnan, N., 2012. The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation,82(8), pp.1191-1206.

Tong, Y.L., 2012. The multivariate normal distribution. Springer Science & Business Media.

Weiss, N.A. and Weiss, C.A., 2012. Introductory statistics. London: Pearson Education.

Yang, Y., Han, S., Wang, T., Tao, W. and Tai, X.C., 2013. Multilayer graph cuts based unsupervised color–texture image segmentation using multivariate mixed student’s t-distribution and regional credibility merging.Pattern Recognition, 46(4), pp.1101-1124.

[1] Hong, Y., 2013. On computing the distribution function for the Poisson binomial distribution. Computational Statistics & Data Analysis, 59, pp.41-51.

[2] Collins, D.J., Neild, A., Liu, A.Q. and Ai, Y., 2015. The Poisson distribution and beyond: methods for microfluidic droplet production and single cell encapsulation. Lab on a Chip, 15(17), pp.3439-3459.

[3] Dai, B., Ding, S. and Wahba, G., 2013. Multivariate bernoulli distribution.Bernoulli, 19(4), pp.1465-1483.

[4] Bueno-Delgado, M.V., Ferrero, R., Gandino, F., Pavon-Marino, P. and Rebaudengo, M., 2013. A geometric distribution reader anti-collision protocol for RFID dense reader environments. IEEE Transactions on Automation Science and Engineering, 10(2), pp.296-306.

[5] Tong, Y.L., 2012. The multivariate normal distribution. Springer Science & Business Media.

[6] Weiss, N.A. and Weiss, C.A., 2012. Introductory statistics. London: Pearson Education.

[7] Yang, Y., Han, S., Wang, T., Tao, W. and Tai, X.C., 2013. Multilayer graph cuts based unsupervised color–texture image segmentation using multivariate mixed student’s t-distribution and regional credibility merging.Pattern Recognition, 46(4), pp.1101-1124.

[8] Ristić, M.M. and Balakrishnan, N., 2012. The gamma-exponentiated exponential distribution. Journal of Statistical Computation and Simulation,82(8), pp.1191-1206.

[9] Krishnamoorthy, K., Mathew, T. and Mukherjee, S., 2012. Normal-based methods for a gamma distribution. Technometrics.

[10] Ben-Israel, A., 2013. A concentrated Cauchy distribution with finite moments. Annals of Operations Research, pp.1-7.