Probability Distribution

A probability distribution is like the perfect or theoretical version of a frequency distribution. A frequency distribution shows how many times each value appears in a real dataset or sample. 

In a sample, the number of times a value appears depends on its likelihood of occurrence. This likelihood is called probability, a number between 0 and 1, where 0 indicates that the event can never happen and 1 indicates that the event is guaranteed.

The closer the probability is to 1, the more often that value will appear in a sample. More precisely, the probability of a value is the relative frequency you would get if the sample were infinitely large. Since we can’t collect infinite data in the real world, probability distributions serve as theoretical models.

They represent the frequency distribution for the entire population. Probability distributions help describe real-world situations, such as coin toss outcomes or the weights of chicken eggs. They are also crucial in hypothesis testing, where they are used to calculate p-values.

Let’s see an example:

If a coin is tossed three times, what will be the probability distribution of the number of heads?
 

Solution:

The possible numbers of heads that can appear are:
(0, 1, 2, 3)
 

There are a total of 23 = 8 possible outcomes when a coin is tossed three times.
 

To get zero heads, the only possible outcome is TTT.
Thus, the probability of getting zero heads = 1/8.
 

To get one head, the possible outcomes are HTT, THT, TTH.
Thus, the probability of getting one head = 3/8.
 

To get two heads, the possible outcomes are HHT, HTH, THH.
Thus, the probability of getting two heads = 3/8.
 

To get three heads, the only possible outcome is HHH.
Thus, the probability of getting three heads = 1/8.
 

The probability distribution is as follows:

 

The two basic forms of probability distributions that describe distinct types of random variables are discrete and continuous probability distributions. It is important to understand their key differences, as this may help in data analysis.

Discrete Probability is utilized when a random variable can take specific, countable values. A discrete random variable may take a countable number of unique values, such as 0, 1, 2, 3...
Discrete probability distributions are used primarily in instances where only a countable number of outcomes can be listed. 

For example: the number of customers visiting a company in an hour, or the probability of getting tails in five coin tosses. 

On the other hand, continuous probability distributions are used to describe continuous random variables, where an infinite set of numbers within a certain range can be applied. These numbers are uncountable, as each interval contains endless possibilities. For example: The exact time required to accomplish a task.

Probability distribution represents how the probabilities of a random variable can be distributed, which helps in predicting the possible outcomes.

The normal distribution, also known as bell curve, is one of the most commonly used probability distributions.

The type of probability distribution a dataset belongs to is based on its data generation process, such as the probability density function (PDF)  and the probability mass function (PMF).

A probability distribution can be shown in a table using a random variable and its possible outcomes.

A random variable X is a rule that assigns a numerical value to each outcome in the sample space of a random experiment.

The probability distribution of X, written as P(X), lists:

• Every possible value that X can take
   
• The probability of each value
   

It can be represented as:

 

Here:

Each probability Pi is greater than zero all the probabilities must add up to 1
P₁ + P₂ + P₃ + … + P_n = 1

Probability distributions are of two types. Let’s learn how they differ:

• Discrete - A discrete probability distribution represents the probability distribution of a random variable that is finite.

• Binomial distribution - Used when an event occurs several times in a set number of trials.
  Each trial has two possible results (such as present or absent, heads or tails).
  Example: The likelihood of getting a head in 10 flips.

• Poisson Distribution - Models the probability of a certain number of events happening within a time or location, assuming the events happen independently at a steady average pace.
  Examples include the number of customers that arrive at a bank each hour and the number of emails received per day.

• Continuous - A continuous probability distribution displays the probability distribution of a random variable that is infinite.

• Normal distribution - A bell-shaped curve is described by its mean and standard deviation.
  It is symmetrical, has no skew, and adheres to the 68-95-99.7% rule (data inside 1, 2, and 3 standard deviations). Example: Individual heights and stock market returns.

Every random variable has a probability distribution that specifies the likelihood of each possible value. Random variables can be:
 

• Discrete – taking specific, countable values
   
• Continuous – taking any value within an interval
   
• Mixed – having both discrete and continuous components
   

A discrete random variable uses a probability mass function (PMF) to show the probability of each possible value.

A continuous random variable is described by a probability density function (PDF) that specifies how probabilities are distributed over an interval of values.
 

Even if two random variables share the same probability distribution, they may still differ in how they relate to other variables, or whether they are independent.
 

When we generate outcomes according to a random variable’s probability distribution, the resulting observed values are called random variates.

Probability distributions help describe how likely different outcomes are in a dataset. Understanding their patterns and practicing applications makes mastering them easier.

• Always start by understanding the basics and distinguishing between discrete and continuous distributions.
   
• Familiarize yourself with key formulas for mean, variance, and standard deviation for each type of distribution.
   
• Visualizing data using graphs and charts helps in understanding how probabilities are spread.
   
• Practice solving real-life problems from finance, games, or experiments to strengthen your understanding.
   
• Compare different distributions under similar conditions to identify patterns and differences.
   

• Use familiar objects like coins, dice, cards, marbles, or candies. Which kids can understand the probability better when they can see the outcomes.
   

• Create a small table showing outcomes and their probabilities. Visual aids help the children quickly understand how the probability distributions work.
   

• Bar graphs, histograms, line plots, and simple charts make the concept easier to grasp. Visuals help learners see patterns, not just memorize formulas.
   

• Simple tools like spreadsheets, apps, or online simulators can show patterns instantly.
  Seeing instant graphs builds better intuition.

We now understand the significance of probability distributions in mathematics. Let’s explore how they play a pivotal role in other fields beyond math:

• Normal distribution is quite often used to predict a student’s performance.

• Students can use the binomial distribution to predict their chances of getting correct answers on tests.

• Binomial distributions are widely used in medical fields to determine the effectiveness of medicines.

• Probability distributions are used in predicting customer behavior based on the trends they follow.

• Poisson distributions play an important role in cybersecurity by detecting fraudulent attempts.