Data distribution PMF, PDF, CDF

Probability Mass Functions (PMFs)

Probability distributions are essential tools in understanding the behavior of random variables. For discrete random variables, we use Probability Mass Functions (PMFs), which assign probabilities to each possible outcome. The sum of these probabilities always equals 1, providing a complete description of the distribution. PMFs allow us to determine the likelihood of observing a specific value of the random variable.

1. PMF of a Fair Six-Sided Die

A fair six-sided die has outcomes {1, 2, 3, 4, 5, 6}. Each outcome has an equal probability of occurring.

$P(X = x) = \begin{cases} \frac{1}{6} & \text{if } x \in \{1, 2, 3, 4, 5, 6\} \\ 0 & \text{otherwise} \end{cases}$

2. PMF of a Biased Coin

A biased coin has a 70% chance of landing on heads (H) and a 30% chance of landing on tails (T).

$P(X = x) = \begin{cases} 0.7 & x = \text{H} \\ 0.3 & x = \text{T} \\ 0 & \text{otherwise} \end{cases}$

Probability Density Functions (PDFs)

Conversely, for continuous random variables, Probability Density Functions (PDFs) are employed. Unlike PMFs, PDFs indicate the likelihood of the variable falling within a particular range. While the PDF itself doesn't provide probabilities directly, the area under the curve within a range represents the probability of the variable falling within that range. Understanding the shape and behavior of PDFs is crucial for analyzing continuous probability distributions.

Uniform Distribution Example:

The time it takes for a bus to arrive at a bus stop, assuming buses arrive at a regular interval. If buses arrive every 10 minutes, and you arrive at the bus stop at a random time, the waiting time $X$ can be modeled as a uniform distribution between 0 and 10 minutes.

​ $f(x) = \begin{cases} \frac{1}{10} & 0 \le x \le 10 \\ 0 & \text{otherwise} \end{cases}$

Cumulative Distribution Function (CDF)

Both PMFs and PDFs are complemented by the Cumulative Distribution Function (CDF). The CDF provides a comprehensive view of the probability distribution by specifying the probability that the variable takes on a value less than or equal to a given value. It accumulates the probabilities of all possible outcomes, starting at zero and approaching one as the variable's value increases. The CDF is indispensable for calculating probabilities, making statistical inferences, and understanding the behavior of random variables across their entire range. By understanding PMFs, PDFs, and CDFs, analysts can effectively model and analyze data in various fields, enabling informed decision-making and predictions.

CDF of a Uniform Distribution

Example: The time you wait for a bus that arrives every 10 minutes uniformly.

The CDF of a uniform distribution over $[a,b]$ is:

​ $F(x) = \begin{cases} 0 & x < a \\ \frac{x - a}{b - a} & a \le x \le b \\ 1 & x > b \end{cases}$

For $a=0 \ and \ b=10$:

$F(x) = \begin{cases} 0 & x < 0 \\ \frac{x}{10} & 0 \le x \le 10 \\ 1 & x > 10 \end{cases}$