What is Probability?

Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.

What is Sample Space?

When any random event takes place that has multiple outcomes, the possible outcomes are grouped together in a set. For any random event or experiment, the set that is formed with all the possible outcomes is called a sample space. The sample space can be anything, from a set of vectors, real numbers, and so on.

What are Random Experiments?

Whenever a test is conducted, whose outcomes are not predictable, the results are termed as random experiments. For instance, when rolling a die once, the resulting number from 1 to 6 cannot be predicted. There can be multiple results, all of which are termed as outcomes. When considering any such outcomes from the sample space, it is called a sample point. With the random experiments, outcomes, and sample points, the pattern of probabilities can be listed. The variables that influence the outcomes are the parameters chosen. These variables are termed random variables.

What is the Probability Distribution of a Random Variable?

A mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment is called a probability distribution. Every random variable comes with its probability distribution. The unknown values also have some probability, which is defined by this term. The random variables can either be continuous, or discrete, or a mixture of both. As a result, a random variable can have any countable or finite value, if it is designated. The probability distribution of the random variable must come with the feature of the probability mass function for this. Otherwise, numerical values in any interval are also possible. When writing a function of the probability distribution, the probability distribution of the random variable is mainly represented. Both continuous and discrete variables can be used together to form a probability distribution of random variables.

However, there can be cases where there is a difference between two random variables having the same probability distribution. This is because the dependency with the other random variables can cause a variation. The term random variates is used to describe the outcomes that come from choosing values randomly, based on the probability distribution function of the variable.

What are the Types of Probability Distribution?

Based on different processes of data generation and multiple purposes, these two classifications have been carried out.

Continuous Probability Distribution

As the name suggests, the values for the different possible outcomes can only be in a continuous range for a continuous probability distribution. In continuous distributions, the graphical representation gives a smooth curve. For instance, the temperature can be a continuous random variable. A continuous probability distribution can be used to estimate the maximum and minimum temperature of a day.

Discrete Probability Distribution

A discrete probability distribution or binomial probability distribution is a distribution where the values for the different possible outcomes can only be of a discrete nature. For instance, while rolling a die, every possible outcome can be said to be discrete in nature. This, in turn, provides a mass outcome. The term probability mass functions are used to describe these masses of outcomes.

Formula

For a Binomial distribution,

$P\left(X=x\right){=}^n{C}_x{g}^x{h}^{n-x}$

Here, ‘g’ and ‘h’ represent the probabilities of success and failure respectively. The random variable for the required event is given by ‘X’, and the number of trials is denoted by ‘n’.

What is Poisson Probability Distribution?

By definition, a Poisson probability distribution can be said to be a special case of the discrete probability distribution. In this distribution, the probability of different events is given. These events must happen in a defined space or time, where the rate and occurrence of each of the events are steady and known with respect to the last event’s occurrence.

Developed by French mathematician Siméon Denis Poisson, it proved extremely useful for determining the probability of events that occur in specified intervals, such as volume or area.

What is Probability Distribution Function?

The function that can define how probabilities of an event have been distributed is called the probability distribution function. There can be various types of probability distribution functions. While considering any random variable, then the probability density functions also get defined with the probability distribution functions.

When considering the normal distribution cases, for a random variable Y with a real value, the function is as follows-

F_y(y) = P (Y ≤ y)

The probability of the occurrence of the random variable Y equal to or less than the value of y is given by P.

What are the Main conditions for any Probability Distribution?

For any probability distribution, the two main conditions are:

• For any random event, the value of the probability will always be within 0 and 1.
• If all the probabilities of the different outcomes are added, then the value should always be 1.

Practice Problem

If a coin is flipped 5 times, calculate the probability of 2 heads.

⇒ Using the equation $P\left(X=x\right){=}^n{C}_x{g}^x{h}^{n-x}$,

Consider n = 5, x = 2.

Also, the success rate and failure rate for a coin are both ½=0.5, as there are only two possible outcomes. Therefore, g = h = 0.5.

Hence, the required probability is

P (X = 2) = ⁵C₂ (0.5)² (0.5)^5-2

⇒ P (X = 2) = ⁵C₂ (0.5)² (0.5)³

⇒ P (X = 2) = 10*(0.25) *(1.25)

⇒ P (X = 2) = 0.3125, which is the required probability.

Context and Application

This topic is significant in the professional exams for both undergraduate and graduate courses, especially for:   

• Bachelor of Science in Mathematics       
• Master of Science in Mathematics