Probability and its Distribution
This is the first article in a 3-part article series.
- The first part will talk about Statistics, Probability, and its distribution curve.
- The second part will talk about commonly used discrete probability distributions including Binomial, Multinomial, Bernoulli, Poisson, and a special case of Uniform distribution.
- The third part will talk about commonly used continuous probability distributions including Normal, Exponential, Chi-square, and a special case of Uniform distribution.
Don’t panic if these terms intimidate or bother you! These articles will try to help you understand these concepts in a friendly way.
Mathematics and Statistics
We all have come around instances where we needed to approximate (guess) a number.
E.g. The distance from the office to home, time taken in climbing up the stairs, number of hours spent watching TV shows, etc.
All these numbers or measures are approximated. It means that we cannot accurately predict these numbers. This is referred to as uncertainty and to account for this uncertainty, we study Statistics.
“Mathematics is certain, while Statistics is Mathematics coupled with uncertainty.”
In statistics, probability plays a significant role in defining a variety of uncertain events. Let us try to understand what is probability and how do we represent it on a graph.
Probability and Density Curves
Suppose we do an experiment that can have any number of outcomes. These experiments can be rolling a dice, flipping a coin, picking a number from 0-10. The idea is that each time you perform the experiment, you are not sure about the results.
The chance of getting a set of the result is called Probability. Therefore, what are the chances of getting a number >3 when we roll a dice?
When we calculate this probability for ALL the possible outcomes, we build a probability distribution for the set of all possible results. The density curve represents this probability distribution.
For example: Suppose we gather information about the weight of 100 employees in an organization. A frequency histogram of this data may look like the below figure:
“The dotted curve represents the density curve”. Now, if we pick any weight range, say 80-85 kg, there will be some employees having those weights. The chance of finding such employees is the Probability. And the chances of finding employees with any possible weight will form the Probability Distribution. This distribution is shown by a relative frequency histogram as below.
Notice how the vertical axis represents the probability in place of the actual number of employees in that weight range. The tallness of the histogram also varies with the weight range of the employees. The taller blocks represent a higher probability of finding an employee in that weight range, while shorter ones explain the opposite.
Hurray! Now you know what probability and its related density curve is. You can also quickly figure out which weight is less probable and which is highly likely by seeing its density curve.
Parameters of a Density Curve:
The behavior of almost every real-world numerical data can be visualized by using a probability distribution. Each of these datasets is usually described by aggregated measures like average, median, mode, standard deviation, etc. These measures provide a one-point summary of the complete data.
Whenever we study the probability distributions, we use such measures to define a distribution. These measures are thus called the parameters of the distribution. The below figure corresponds to a Normal Distribution with its parameters being Mean and Standard Deviation.
The third part of this article series will talk about the Normal Distribution in detail.
Types of Probability Distributions
Based upon the nature of data, probability distributions can be:
- Discrete Probability distribution
- Continuous Probability distribution
Discrete Probability Distribution: It is used for data having discrete or individual outcomes, such as getting a Head or a Tail, or getting a number between 1-6 after rolling a dice, etc.
Continuous Probability Distribution: It is used in cases where the data is collected on a continuous scale, such as the height of trees in a forest, the weight of employees in an organization, etc.
We will study in great detail about some of the commonly used probability distributions in Part 2 and 3.