"INTRODUCTIONStatistical distributions and models are commonly used in many applied areas such aseconomics, engineering, social, health, and biological sciences. In this era of inexpensive andfaster personal computers, practitioners of statistics and scientists in various disciplines have nodifficulty in fitting a probability model to describe the distribution of a real-life data set. Indeed,statistical distributions are used to model a wide range of practical problems, from modeling thesize grade distribution of onions to modeling global positioning data. Successful applications ofthese probability models require a thorough understanding of the theory and familiarity with thepractical situations where some distributions can be postulated.Certain probability distributions occur with such regularity in real-life applications that they havebeen given their own names. Here, in this project we will survey and study basic properties ofStatistical distribution. 1. PROBABILITY DISTRIBUTION:In probability and statistics, a probability distribution is amathematical function that, stated in simple terms, can be thought of as providing the probabilityof occurrence of different possible outcomes in an experiment. For instance, if the randomvariable X is used to denote the outcome of a coin toss ('the experiment'), then the probabilitydistribution of X would take the value 0.5 for = ? ? ? ? ? , and 0.5 for ? = ? ? ? ? ?In more technical terms, the probability distribution is a description of a random phenomenonin terms of the probabilities of events. Examples of random phenomena can include the result ofan experiment or survey.Probability distributions are generally divided into two classes. ? Discrete probability distribution? Continuous probability distribution1.1 Discrete probability distribution: Applicable to the scenario where the set of possible outcomes is discrete, such as acoin toss or a roll of dice) can be encoded by a discrete list of the probabilities of theoutcomes, known as a probability mass function. 1.2 Continuous probability distribution: Applicable to the scenarios where the set of possible outcomes can take on values in acontinuous range (e.g., real numbers), such as the temperature on a given day) istypically described by probability density functions (with the probability of anyindividual outcome actually being 0.We will discuss the following distributions:• Binomial• Poisson• Uniform• NormalThe first two are discrete and the last two are continuous.1.1.1Binomial Distribution:History: In 1713 Swiss mathematician Jakob Bernoulli, determined that the probability of ksuch outcomes in n repetitions is equal to the kth term (where k starts with 0) in the expansion ofn the binomial expression (p + q) , where q = 1 - p.In the example of the die, the probability of turning up any number on each roll is 1 out of 6 (thenumber of faces on the die). The probability of turning up 10 sixes in 50 rolls, then, is equal to50 the 10th term (starting with the 0th term) in the expansion of (5/6 + 1/6) , or 0.115586.In 1936 the British statistician Ronald Fisher used to publish evidence of possible scientificchicanery-in the famous experiments on pea genetics reported by the Austrian botanist GregorMendel in 1866. Fisher observed that Mendel’s laws of inheritance would dictate that the3 number of yellow peas in one of Mendel’s experiments would have a with n = 8,023 and p = / ,4 for an average of np ? 6,017 yellow peas. Fisher found remarkable agreement between thisnumber and Mendel’s data, which showed 6,022 yellow peas out of 8,023. One would expect thenumber to be close, but a figure that close should occur only 1 in 10 times. Fisher found, moreover, that all seven results in Mendel’s pea experiments were extremely close to theexpected values.Assumption for Binomial Distribution:There is a set of assumptions which, if valid, would lead to a binomial distribution. These are:? A set of n experiments or trials are conducted.?Each trial could result in either a success or a failure.?The probability p of success is the same for all trials.?The outcomes of different trials are independent.?We are interested in the total number of successes in these n trials.Under the above assumptions, let X be the total number of successes. Then, X is called abinomial random variable, and the probability distribution of X is called the binomialdistribution.Binomial Probability-Mass Function:Let X be a binomial random variable. Then, its probability-mass function is:? ? ? -? ? ? = ? =? 1-? ? Where n and p are called the parameters of the distribution.Applications of Binomial Distribution:All of these are situations where the binomial distribution may be applicable.? The number of heads/tails in a sequence of coin flips.? Vote counts for two different candidates in an election.?The number of male/female employees in a company. ?The number of accounts that are in compliance or not in compliance with an accountingprocedure.?The number of successful sales calls.?The numbers of defective products in a production run.?The number of days in a month your company’s computer network experiences a problem.1.1.2Poisson Distribution:History: In probability theory and statistics, the Poisson distribution; named after Frenchmathematician Siméon Denis Poisson, is a discrete probability distribution that expresses theprobability of a given number of events occurring in a fixed interval of time and/or space if theseevents occur with a known average rate and independently of the time since the last event. ThePoisson distribution is another family of distributions that arises in a great number of businesssituations. It usually is applicable in situations where random “events” occur at a certain rateover a period of time.Poisson Probability-Mass Function:Let X be a Poisson random variable. Then, its probability mass function is:? ? -?? ? = ? = ?? ? ? ? = 0,1,2,3,…? ! The value of µ is the parameter of the distributionApplications of Poisson distribution:Consider the following scenarios: ? The hourly number of customers arriving at a bank.?The daily number of accidents on a particular stretch of highway. "