A Beginner’s Guide to Understanding Probability Density and Mass Functions
Navigating the Landscape of Probability: Demystifying PDF, CDF, PMF, and CMF
Introduction
Probability is a fundamental concept that underpins many fields, from statistics to machine learning. To better understand the distribution of random variables and the likelihood of certain outcomes, we use mathematical functions called Probability Density Functions (PDF), Cumulative Density Functions (CDF), Probability Mass Functions (PMF), and Cumulative Mass Functions (CMF).
In this article, we’ll break down these concepts in an easy-to-understand manner and explore the relationships between them.
Probability Functions
1. Probability Mass Function (PMF)
Imagine flipping a fair coin. The PMF gives us the probability of getting each possible outcome. For a discrete random variable, like the result of a coin flip, the PMF assigns probabilities to specific values. Let’s denote the random variable as X and the value it can take as x. The PMF is usually denoted as P(X = x).
In a dice-rolling example, the probability of getting one outcome would be 1/6 since the outcomes would be independent.
2. Cumulative Mass Function (CMF)
The CMF gives us the cumulative probability of a random variable being less than or equal to a certain value. Mathematically, it is denoted as F(X = x) and calculated as the sum of probabilities up to x.
Using the coin flip scenario, the CMF at x = 1 (heads) would be 0.5, and at x = 2 (tails), it would be 1.0.
Notice how the probability is adding up. Let’s say if I want to get the Probability of getting 5 or less than 5 I would have to sum up all the probabilities meaning I would get 5/6. As the name suggests, the probability starts adding up.
3. Probability Density Function (PDF)
Consider measuring the height of individuals in a population. Unlike coin flips, this is a continuous random variable. The PDF provides the probability of the random variable falling within a certain range.
A continuous random variable is a type of random variable that can take any value within a certain range or interval. Unlike discrete random variables, which can only take on distinct, separate values (like integers), continuous random variables can take on an infinite number of possible values within a specified range.
For a continuous random variable X, such as the heights of individuals, and a height range [a, b], the PDF is denoted as f(X) and satisfies the property that the integral of f(X) over the height interval [a, b] equals the probability of X (height) falling within that range.
In the height example, the PDF could show that heights between 160 cm and 170 cm have a higher probability.
So if I want to get the probability of a person having a height equal to 165 cm, it would be 0.04.
4. Cumulative Density Function (CDF)
The CDF for continuous random variables gives us the probability that the random variable X is less than or equal to a specific value x. It is denoted as F(X = x) and is calculated by integrating the PDF up to x.
In the height scenario, the CDF at x = 165 cm would indicate the cumulative probability of individuals having a height less than or equal to 165 cm.
Okay but how would I get the PDF given I have the CDF available? I would go ahead and take the point at which I want to calculate the probability, go a step behind and a step further and take the X values and then get the probability of those two points. Then I would go ahead and find out the derivative to then get the probability value at a certain height.
I would get the derivative by differencing the values in y and dividing it by the difference in the values of x. By that logic, I would get 0.04, and voila!
Relationships Between Functions
The relationships between these functions are intuitive yet crucial. The PMF and CMF are discrete counterparts of the continuous PDF and CDF. The CDF is the integral of the PDF, while the CMF is a sum of probabilities from the PMF.
In simple terms, the CDF and CMF provide a broader view of probabilities by considering ranges of values, whereas the PDF and PMF focus on individual values.
Conclusion:
1. Ability to measure uncertainty: Probability Functions as Foundational Tools Statistics, data analysis, and machine learning all rely on probability functions, such as PMF, CMF, PDF, and CDF. They give us the ability to measure uncertainty and obtain an understanding of the probability of various possibilities.
2. Wide Range of Applications: These ideas are widely used in a number of different fields. Probability functions offer an organized approach to comprehending and dealing with randomness, from forecasting the results of coin tosses to modeling complicated real-world phenomena like human heights.
3. Complementary Nature: These functions have interwoven linkages. Discrete scenarios are handled by PMF and CMF, whilst continuous scenarios are handled by PDF and CDF. A comprehensive perspective of cumulative probabilities is provided by CDF, which is obtained from the integral of PDF, whereas CMF combines probabilities from the PMF.
4. Maintaining a Balance Between Individual and Aggregate Probabilities: While CDF and CMF capture probabilities throughout ranges, PDF and PMF concentrate on single values or events. We may balance a thorough analysis of individual outcomes with a more comprehensive comprehension of aggregate probabilities by understanding how these functions interact.
5. Improved Data Interpretation: Possessing mastery over these probability functions equips people to conduct more insightful analyses of data. A firm understanding of these ideas is a great advantage whether making decisions based on complex statistical patterns or drawing conclusions from large datasets.
In summary, the study of probability functions offers a methodical framework for addressing uncertainty and estimating randomness.
To do this, we must first understand the complexities of probability density functions, cumulative density functions, probability mass functions, and cumulative mass functions.
Happy Learning!!
