Discrete Probability Distributions - Essay Example

Name of Student) (Course Instructor) (Course title) (Date of submission) Summary Assignment Discrete Probability Distributions Phenomena which are subject to random influences normally have results that are random in nature. In other words, the output is called as random variables. To be able to represent such situations probability distribution is applied. In discrete probability distributions values of a random variable are presented in form of a list. This is different from frequency distributions where probability distribution may be illustrated in form of mean and variance (Engineering Mathematics 1). The discussion in chapter three of the course is about discrete probability distributions and various concepts are explained including random variables, expected value and variance, binomial probability distribution and Poisson probability distribution. Random Variables The informal definition of a random variable is that, it is a value of a measurement that results out of an experiment (Busken 1). For example, the numbers of heads that would appear when a coin is tossed n times explain a random variable. However, formally a random variable may be defined as a function that maps that maps each and every sample outcome to a real number. In simple terms, a random variable is description of the outcome of an experiment in numerical terms (Thomson South-Western 2). According to Education Service Australia (5) a variable is made random because a value of a random variable cannot be observed until the random procedure is performed. There are two distinct types of random variables; discrete random variable and continuous random variable. These variables are handled differently. Discrete random variables exist in form of finite numbers of values or may take the form of infinite sequence of values. In other words, they take values restricted to different separate values (Busken 1). On the other hand, continuous random variables assume any numerical value either in a single interval or many intervals. We can use JSL TV to illustrate the concept of discrete random variable, both with finite and infinite numbers of values. First, let’s consider discrete random variable having a finite number of values. Consider (x) to be number of JSL TVs sold by the store keeper in one day. In this case, (x) can assume five values, such as 0, 1, 2, 3, 4 (Thomson South-Western 4). This means that in any single day, a store keeper may sell from 0 TVs to 4 TVs. There is no way, for instance, 3.5 TVs can be sold by the store keeper. Therefore, random variables that assume such values are known as discrete random variables with a finite number values. Also, consider (x) to be number of customers who visit the store to but JSL TV set, where (x) can assume the values 0, 1, 2, 3, 4, 5,….In this case, it is possible to count the number of customers visiting the store in the course of the day (Thomson South-Western 4). However, we cannot exactly tell the number of customers who are likely to arrive at the store in one day. In other words, there is no finite upper limit on the number that is likely to arrive. The customers arriving at the store might be as many as 10, 20, or even more than 30. This illustration explains discrete random variable with an infinite sequence of values. Generally, various questions can be used to determine whether a random variable is classified as discrete random variable and continuous random variable. For example, a question may be based on family size where an individual could be interested in knowing the number of dependents (x) in family who are incorporated on tax return (Thomson South-Western 4). This type of random variable is discrete in nature. The same applies to a question asked to a person whether he or she own dog of cat, where our random variable is represented by (x). Thus, (x) may assume values 1, 2, 3, 4 representing owning no pet, owning dog (s) only, owning cat (s) only, and owning dog (s) and cat (s) respectively (Thomson South-Western 4). This type of information is in form of discrete random variable as well. Nevertheless, a question about distance taken by an individual from one place another yields continuous type of random variable. For example, the distance (x) in miles, a store keeper would cover from home to the site of the store produces values associated with continuous random variable (Thomson South-Western 4). Discrete Probability Distributions Another concept that emerges from the study of discrete probability distributions is what is referred to as probability density functions. They are normally used to describe how random variables are distributed over the values of the random variable, also called probability distribution (Thomson South-Western 6). Discrete probability distribution can be described by use of a table, graph, or equation depending on what a person considers to be appropriate. A probability function helps to define the probability distribution, where by the probability function is denoted by f(x). The probability function gives the probability for every value of the random variable. Two conditions are required for a discrete probability function and they are denoted as f(x) ≥ 0 and ∑ f(x) = 1(Thomson South-Western 7). One of the simplest examples of discrete probability distribution is referred to as discrete uniform probability distribution. It is represented in a formula form as follows: f(x) = 1/n where n represents the number of values likely to be assumed by the random variable. Expected Value and Variance Expected value and variance are also important elements probability distribution. The expected variance is also known as mean and is used to explain a measure central location of a random variable (Thomson South-Western 11). It is denoted as follows: E(x) = u =∑ xf (x) On the other hand, variance is used to summarize the differences in the values of a random variable (Thomson South-Western 11). It is denoted as follows: Var (x) = σ2 = ∑(x- u)2f(x) The standard deviation is represented as σ and it is the positive square root of the variance (Thomson South-Western 11). Binomial Probability Distribution According to Thomson South-Western (16) a binomial distribution provides probabilities that are closely associated with independent Bernoulli trials that are repeated. A Binomial experiment has four distinct properties which must be understood. The first property is that the experiment involves a series of n identical trials. Second, the experiment is associated with two outcomes, that is, success and failure, which have the same probability to occur on every trial. Third, the probability of an outcome of success (þ) is stationary from trial to trial. Lastly, the trials are independent, that is, they do not compete with each other. This paper focuses on the number of success that takes place in the n trials. In this case we shall denote the number of success occurring in a trial by (x). Therefore, the binomial probability function is shown as follows: f (x) = n! px(1-p)(n-x) x!(n-x)! Where: fx = probability of x success in n trials n = number of trails p = probability of success on any trial Illustration There has been a low level of retention in the company where Evans works in the past years and this has made him more concerned about the situation. Consequently, a 10% turnover of the hourly employees per year has been noticed by management. This means whenever an hourly employee is selected randomly, it can be estimated with a probability of 0.1% that the employee will not still working in the company the following year (Thomson South-Western 20). For example, when management chooses 3 hourly employees randomly, what is the likelihood that 1 of them will not be working with the company by the end of next year, when p = .10, n = 3 and x =1. Then using the binomial probability function, it can be substituted as follow: f (1) = 3! (0.1)1(0.9)2 =3(.1) (.81) = .243 1!(3-1)! We can also compute the expected value, variance and standard deviation using the illustration above. This can be done as follows: Given the formulas Expected Value E(x) = u = np Variance Var (x) = σ2 = np(1-p) Standard deviation σ = Then: Expected value = E(x) = u = 3(.1) = 3 employees out of 3 Variance = Var (x) = σ2 = 3(.1) (.9) = .27 Standard deviation = = .52 employees Poisson Probability Distribution A random variable which is Poisson distributed is used to estimate the number of occurrences over a known interval of time. Poisson probability distribution is an a discrete random variable likely to assume an infinite sequence of values 0, 1, 2,……(Thomson South-Western 29). Examples of such variable could be the number of cars arriving at a toll booth in 30 minutes or the number of knotholes in 20 linear feet of a pine board. A Poisson experiment has two properties. The first one is that the probability of an occurrence taking place is similar for any two intervals having the same length. The second property is that either the occurrence or non-occurrence of the probability in one interval is independent of the occurrences of non-occurrences that take place in another interval (Thomson South-Western 32). Poisson probability function is illustrated as below: f (x) = ux e- u/x! Where; fx = probability of x Occurrence in an interval u = mean number of occurrence in an interval e = 2.71828 Works Cited Busken, Tim. "Discrete Probability Distributions." Professor Tim Busken (2013). Education Services Australia. “Discrete probability distributions – A guide for teachers (Years 11–12),2013. Engineering Mathematics. “Discrete Probability Distribution,” 2005 Thomsom South-Western. “Discrete Probability Distributions,”2006 Read More