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

Probability Distribution Function (PDF) for Discrete Random Variables

Probability calculation for discrete variables

A probability distribution function (PDF) for a discrete random variable $X$ assigns a probability to each possible value $X$ can take. You write it as $P(X = x)$, where $x$ is one specific outcome.

Think of it as a complete table that tells you: "Here are all the things that could happen, and here's how likely each one is."

To calculate probabilities using a PDF:

1. List every possible value of $X$ (for a standard die, that's 1, 2, 3, 4, 5, 6).
2. Look up the probability assigned to each value (for a fair die, $P(X = 1) = 1/6$).
3. If an event covers multiple values, add their individual probabilities together.

For example, suppose $X$ can only take values 1, 2, or 3 with these probabilities:

• $P(X = 1) = 0.2$
• $P(X = 2) = 0.3$
• $P(X = 3) = 0.5$

To find $P(X \leq 2)$, you add the probabilities for every value that satisfies $X \leq 2$:

$P(X \leq 2) = P(X = 1) + P(X = 2) = 0.2 + 0.3 = 0.5$

This works because the values of a discrete random variable don't overlap, so you can always combine probabilities by adding.

Validation of probability distribution functions

Not every table of numbers qualifies as a legitimate PDF. A valid PDF must satisfy exactly two conditions:

• Non-negativity: $P(X = x) \geq 0$ for every possible value of $x$. Probabilities can never be negative.
• Probabilities sum to 1: $\sum_{x} P(X = x) = 1$. When you add up the probabilities across all possible values, the total must be exactly 1 (since something has to happen).

To check whether a given PDF is valid:

1. Confirm that every listed probability is zero or positive.
2. Add all the probabilities together and verify the sum equals 1.

If either condition fails, you're not looking at a valid probability distribution. For instance, if someone hands you a table where the probabilities add to 0.95, that's not a valid PDF because 5% of the probability is unaccounted for.

Interpretation of discrete probabilities

The numbers in a PDF aren't just abstract values. They connect to real outcomes in two ways:

• Long-run frequency: Over many repeated trials, the proportion of times you observe a particular outcome will approach its PDF probability.
• Single-trial likelihood: For any one trial, the probability tells you the chance of that specific outcome occurring.

Suppose a PDF models the number of defective items per batch on a production line, and $P(X = 2) = 0.1$. That means:

• If you inspect many batches over time, about 10% of them will contain exactly 2 defective items.
• For any single batch, there's a 0.1 (10%) chance it contains exactly 2 defective items.

Both interpretations say the same thing from different angles. The long-run interpretation is supported by the law of large numbers, which says that as you observe more and more trials, the observed proportions get closer and closer to the true probabilities.

Additional Probability Concepts

A few related ideas that often come up alongside PDFs:

• Independence: Two events are independent if the occurrence of one doesn't change the probability of the other. Rolling a 4 on one die doesn't affect what you roll on a second die.
• Conditional probability: The probability of an event occurring given that some other event has already happened. This matters when events are not independent.
• Random sampling: Selecting individuals from a population so that every member has an equal chance of being chosen. This helps ensure the sample is unbiased and that probability models apply properly to the data you collect.