4.1 Probability Distribution Function (PDF) for a Discrete Random Variable

Probability Distribution Functions (PDFs) for discrete random variables are key tools in statistics. They let you calculate the likelihood of specific outcomes in scenarios with countable possibilities, like rolling dice or counting defective items in a shipment.

Understanding PDFs is crucial for analyzing real-world data and making predictions. This section covers how to use these functions, how to distinguish between discrete and continuous variables, and how to verify whether a given function qualifies as a valid probability distribution.

Probability Distribution Function (PDF) for Discrete Random Variables

Fundamental Concepts

Before diving into PDFs, you need a few building blocks:

• Sample space: The set of all possible outcomes of a random experiment. For a single die roll, the sample space is {1, 2, 3, 4, 5, 6}.
• Probability axioms: The fundamental rules that all probabilities must follow. Every probability is between 0 and 1, and the probabilities of all outcomes in a sample space sum to 1.
• Independence: Two events are independent when the occurrence of one does not affect the probability of the other. Drawing two cards with replacement gives independent events; drawing without replacement does not.
• Conditional probability: The probability of an event occurring given that another event has already occurred, written $P(A \mid B)$.

Probability Calculations for Discrete Variables

A random variable is a function that assigns a numerical value to each outcome in the sample space. For example, if you roll two dice, you could define $X$ as the sum of the two faces.

The Probability Distribution Function (PDF) for a discrete random variable $X$, denoted $f(x)$, gives the probability that $X$ takes on a specific value $x$:

$f(x) = P(X = x)$

(You'll also see this called a probability mass function, or PMF. For discrete variables, the two terms refer to the same thing.)

To calculate probabilities using a PDF:

1. Identify the random variable $X$ and list its possible values.
2. Find the corresponding probability for each value using the given PDF.
3. For the probability of multiple values, add the individual probabilities together. This works because discrete outcomes are mutually exclusive.

Example: Suppose $X$ has the following PDF:

• $P(X = 2) = f(2) = 0.3$
• $P(X = 1 \text{ or } X = 3) = f(1) + f(3) = 0.2 + 0.5 = 0.7$
• $P(X \leq 2) = f(1) + f(2) = 0.2 + 0.3 = 0.5$

Discrete vs. Continuous Random Variables

This distinction matters because the type of variable determines how you find probabilities.

Discrete random variables take on a countable number of distinct values. They typically arise from counting: the number of defective items in a batch, the number of customers in a queue, or the number of heads in 10 coin flips. You find probabilities by summing the values of the PMF. A special case is the discrete uniform distribution, where every outcome has equal probability (like a fair die, where each face has probability $\frac{1}{6}$).

Continuous random variables take on an uncountable number of values within a range. They typically arise from measuring: the weight of a product, the time until a machine fails, or a person's height. Probabilities are found by integrating the density function over an interval, not by summing individual values. For a continuous variable, $P(X = x) = 0$ for any single point, which is a major conceptual difference from the discrete case.

Validity of Discrete Probability Distributions

Not every function qualifies as a valid PDF. A function $f(x)$ is a valid discrete probability distribution only if it satisfies two conditions:

1. Non-negativity: $f(x) \geq 0$ for all values of $x$. Probabilities can never be negative.
2. Normalization: $\sum_{\text{all } x} f(x) = 1$. The probabilities across all possible values must sum to exactly 1.

The function should also only assign nonzero probabilities to the actual possible values of the random variable (its support). Any value outside the support has $f(x) = 0$.

To verify a distribution is valid:

1. Check that every listed probability is non-negative.
2. Add up all the probabilities and confirm the total equals 1.
3. Confirm that no probability is assigned to values outside the variable's possible range.

Example: Is this a valid distribution?

• All values are $\geq 0$. ✓
• $0.1 + 0.3 + 0.4 + 0.2 = 1.0$. ✓

This is a valid distribution. If the sum had been anything other than 1, or if any value had been negative, it would fail.