11. Probability Distributions (Discrete)

• Outcome X takes values x₁, x₂, … xₙ with probabilities p₁, p₂, … pₙ.

In this context, a discrete random variable X is one that can take on a finite or countably infinite number of separate values. For example, when you roll a die, the outcome (let's say X) can be 1, 2, 3, 4, 5, or 6. Each of these outcomes corresponds to a probability, denoted p₁, p₂, ..., and so on, where p represents the chance of each value occurring.

Think of a jar filled with colored marbles. If you take one marble out at random, the possible outcomes are the colors of the marbles present (like red, blue, green, etc.), and each color has a specific chance of being picked based on the total number of marbles.

• Must satisfy: ∑pᵢ = 1 and 0 ≤ pᵢ ≤ 1.

The probabilities associated with the outcomes of a discrete random variable must meet two crucial conditions: the sum of the probabilities of all possible outcomes must equal 1. This means that when you consider all possible events, something must happen. Additionally, each individual probability must be between 0 and 1, meaning it can't be less than impossible (0) or more than certain (1).

If we return to our marble jar, if we know there are 10 marbles total, and 3 are red, then the probability of pulling a red marble is 3/10 (which is 0.3). If you sum the probabilities of all colors (like red, blue, green), they should add up to 1, meaning there's a complete certainty that you will pull one color, no matter which one it is.

• Expected value (mean): 𝐸(𝑋) = ∑𝑥 𝑝 .

The expected value, or mean, of a discrete random variable X is calculated by multiplying each possible outcome by its probability and then summing all these products. This measure provides a single value that represents the average or 'center' of the distribution of outcomes—basically, it is what you would expect to happen on average over many trials.

Imagine you're playing a game where if you roll a die you earn a number of points equal to the number rolled. If you calculate the expected number of points you will earn from rolling the die repeatedly, you'd take each outcome (1 through 6), multiply by its probability (1/6), and find that the expected value is 3.5 points.

• Variance: 𝑉𝑎𝑟(𝑋) = ∑(𝑥 −𝐸(𝑋)) 𝑝 .

Variance measures the spread or dispersion of a set of outcomes of a random variable. It is the average of the squared differences from the expected value. A high variance indicates that the outcomes are spread out widely around the mean, while a low variance indicates that they are clustered closely around the mean.

Continuing with our die-rolling game, if the variance in the points earned is low, this means your scores when rolling the die won't fluctuate much—you might consistently score around 3.5 points. However, if the variance is high, some rolls could yield very low or very high scores, making the game unpredictable.

• Example: Rolling a fair die → E(X)=3.5, Var(X)=35/12.

In the example of rolling a fair die, we can calculate both the expected value and variance. The expected value E(X) is calculated based on the outcomes {1, 2, 3, 4, 5, 6} and their equal probabilities (1/6). This gives us an average outcome of 3.5. The variance can be computed similarly using the formula provided, leading to a numerical value representing how far the outcomes typically deviate from the mean.

Imagine a group of friends playing a board game that uses a fair die. If they record their scores from multiple rounds, they’ll find that over many games, their average score tends to even out to 3.5, and some scores occasionally float far above or below this average, as reflected in the variance of their rolls.

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