Frequently, a number is organically related to the outcome of a random experiment: the number of boys in a family with three children, the number of defective light bulbs in a case of 100 bulbs, the amount of time until the next client comes at the bank’s drive-up window. This number varies from trial to trial of the relevant experiment in an unpredictable manner; therefore, it is referred to as a random variable. In this and the following chapter, we will examine such variables.
Types of Random Variables
As mentioned in the introduction, there are two random variables, including:
Independent Random Variable
Random Continuous Variable
Discrete Random Variables
A discrete random variable may assume either a finite or countably infinite set of discrete values (for example, the integers). Their probability distribution is determined by a probability mass function that explicitly associates each random variable value with a probability. For instance, the value of x1 is assigned the probability p1, the value of x2 is assigned the probability p2, etc. The probabilities pi must meet two conditions: each probability must be between 0 and 1, and the sum of all probabilities must equal 1. (p1+p2+⋯+pk=1)
Examples of discrete random variables include the outcomes of a die roll and test scores out of a possible 100.
Continuous Random Variables
Continuous random variables, on the other hand, have values that fluctuate continuously within one or more real intervals and possess a continuous cumulative distribution function (CDF). As a result, the random variable has an uncountable infinite number of possible values, all of which have a probability of zero, although ranges of such values can have probabilities greater than zero. The resultant probability distribution of the random variable can be represented by a probability density, where the probability is determined by calculating the area under the curve.
There are an endless amount of alternatives for continuous random variables, such as selecting random values between 0 and 1.
Probability Distributions for Discrete Random Variables
Probability distributions for discrete random variables may be represented as an equation, a table, or a graph.
A discrete probability function must meet the aforementioned conditions: 0≤f(x)≤10≤f(x)≤1, i.e., f(x) values are probabilities, therefore between 0 and 1.
A discrete probability function must also meet the aforementioned conditions: ∑f(x)=1 f∑ (x) = 1, i.e., by summing the probabilities of all disjoint cases, we obtain the sample space probability of 1.
The probability mass function serves the same goal as the probability histogram, displaying probabilities specific to each discrete random variable. The sole distinction is in its visual appearance.
Discrete random variable: produced by counting discrete values, such as the integers 0, 1, 2,…
Probability distribution: A function of a discrete random variable that yields the likelihood that the variable will have a particular value.
Probability mass function: a function that provides the relative likelihood that a discrete random variable is equal to a certain value
A discrete random variable x can take on a finite number of distinct values. The probability distribution of a discrete random variable xx specifies the values and their probabilities, such that value x1 has probability p1, value x2 has probability x2, etc. Each probability pi is a positive integer between 0 and 1, and the sum of all probabilities equals 1.
Suppose, for instance, that x, a random variable representing the number of persons waiting in line at a fast-food restaurant, can only take on the values 2, 3, or 5 with a probability of 2/10, 3/10, and 5/10, respectively. This can be stated using the function f(x)=x/10, with x = 2, 3, 5, or the table below. Of the conditional probability of the occurrence B given the conditions A1 and A2, respectively. Observe that these two representations are equivalent and that this is graphically expressed in the probability histogram below.
This probability histogram illustrates the probabilities of the three discrete random variables.
The formula, table, and probability histogram satisfy the requisite conditions for discrete probability distributions:
0≤f(x)≤10≤f(x)≤1, i.e., f(x)f(x) values are probabilities, therefore between 0 and 1.
∑f(x)=1∑f(x)=1, i.e., by combining the probabilities of all disjoint cases, we obtain the sample space probability, which is 1.
The discrete probability distribution is occasionally referred to as the probability mass function (pmf). The probability mass function serves the same role as the probability histogram, displaying probabilities for every discrete random variable. The sole distinction is in its visual appearance.
Values Expected for Discrete Random Variables
The expected value of a random variable is the weighted mean of all its possible values.
The expected value of a random variable X is defined as E[X]=x1p1+x2p2+⋯+xipi, which can also be written as: E[X]=∑xipi.
If all outcomes xi have the same probability (p1=p2=…=pi), the weighted average becomes the simple average.
The expected value of X is what one expects to occur on average, even though it occasionally results in an impossible figure (such as 2.5 children).
The following are examples of discrete random variables:
The number of eggs laid each day by a hen (it cannot be 2.3).
The number of individuals attending a specific soccer match
The number of pupils attending class on a certain day
The number of individuals waiting in line at McDonald’s on a certain day and time
The following are 10 examples of discrete random variables:
The number of possible outcomes of a fair coin flip.
The number of students present in the classroom.
The number of awards received during the academic year.
The daily incidence rate of covid cases.
The number of patients within a ward.
The number of vaccination dosages.
The number of eggs sold per day.
The number of Novel Corona Virus 19 recoveries every week.
The number of equations utilised to solve a problem.
The number of questions on the examination.
The outcome of a random experiment is quantified using a discrete random variable. Discrete Random Variable has a countable quantity of outcomes. In general, discrete random variables can be categorised as 0, 1, 2, 3, 4,…… The data may be discrete or continuous, however, we will only explore discrete random variables here. Probability distributions are utilised to illustrate the distribution of probabilities over the values of discrete random variables.