Binomial Random Variables

What are binomial random variables?

A variable is said to be a binomial random variable if the outcome of a random experiment decides its value. A random variable is capable of taking on any real value.

A random variable is a real-valued function whose domain is a random experiment’s sample space S and whose value is the experiment’s result. A random variable is often represented by a capital letter, such as X, Y, or M. Lowercase letters such as x, y, z, m, and so on, denoting the random variable’s value.

Consider the random experiment of tossing a coin three times. If you get heads, you receive Rs. 5, and if you get the tails, you lose Rs. 5. You and your friend are ready to battle to see who can make the most money and win the game by tossing 3 times. We can see that the likelihood of getting a head for the coin after three throws vary from zero to three. If the letter X represents the number of heads, then

X = {0,1,2, 3} where 0 is the initial integer. The probability of always acquiring a head is 1/2.

Types of random variables

There are several different kinds of random variables.

• Discrete random variable

In keeping with the name, this metric is not linked or continuous. A discrete variable has only a finite number of possible values, such as a discrete random sample. There is a chance that the random variable will have a value. Distinct random variables have real-valued functions defined on a discrete sample space.

Distinct random variables include the number of calls a person receives in a day, the number of items sold by a company, the number of items manufactured and so on.

• Continuous random variable

A continuous random variable is one that assumes the sample space has infinite values. It is capable of accepting any value between a set of predetermined limits. Integral and fractional values can also be entered. Continuous random variables include a person’s height, weight, age, and the distance between two cities.

Types of probability distribution

Probability distributions are classified into two types, each used for a different purpose and in a different kind of data generation process than the others. Probability distributions are classified into two types: normal and cumulative. Binomial and discrete probability distributions are the two types of probability distributions.

Probability distribution function

General statistics define the probability of a favourable result as the ratio of favourable outcomes to the total number of events in a sample space. It is written mathematically as follows: P(E) = (Number of good outcomes) x Probability of occurrence (Sample space).

The distribution of possibilities 

For each event in a random experiment, the chance of every recurrence may be computed. We can determine the probability of a specific random variable value for various random variable values. A random variable’s probability distribution comprises the values of random variables and the probabilities associated with each of those values.

Consider the situation in which X is a random variable. The function P represents the probability distribution of a variable X. The distribution function of the random variable X is defined as any function F defined for all real x and determined by the equation F(X) = P(X).

Properties of random variables

• Due to random outcomes, the random variable takes more values. It is not similar to a case of algebraic variable
• As algebraic variables represent the unknown values given in the algebraic equation
• Her in case of random variables represent all short of possible outcomes during an experiment
• The variable whose values depends upon the numerical outcome of certain random phenomena is called a random variable
• The other name of the random variable is the Stochastic variable
• The numerical values of the random variable are real numbers and in measurable quantity
• The discrete variable and continuous random variable is described by probability mass function and probability density function, respectively

Conclusion

As we’ve seen, a variable is anything whose value may fluctuate. It might change depending on the results of an experiment. When a random experiment determines the value of a given variable, it is called a random variable. A random variable may have any actual value at any time.

An experimental sample space S defines the domain of a random variable, which is a real-valued function. X, Y, M, and other capital letters signify random variables. The random variable’s value is denoted by lowercase letters like x, y, z, m, and so on.