Chapter 6: Discrete Probability Distributions

A random variable is defined as a numerical measure of the outcome from a probability experiment, determined by chance. Random variables are typically denoted with letters such as $X$.
A discrete random variable has either a finite or countable number of values. Values can be plotted on a number line with gaps between points.
- Example 1: The number of A's earned in a statistics class with 15 students. Possible values: $x = 0, 1, 2,
  ightarrow 15$.
- Example 2: The number of cars that pass through a McDonald's drive-through in the next hour, where possible values are $x = 0, 1, 2,
  ightarrow ext{(unbounded)}$.
A continuous random variable has infinitely many possible values. These values can be plotted in an uninterrupted manner on a line.
- Example 3: The speed of a car passing a state trooper falls under continuous variables, represented mathematically by the inequality $s > 0$ (all positive real numbers).

The probability distribution of a discrete random variable $X$ provides the possible values of $X$ and their corresponding probabilities. This can be represented in tables, graphs, or mathematical formulas.
Example: Consider a basketball player making three free throws, where $x = 0, 1, 2, 3$.
- Notation: Probability $P(x)$ is the probability that random variable $X$ equals the value $x$. For example, $P(3) = 0.51$ means the probability of making 3 shots is 0.51.
Rules for a Discrete Probability Distribution:
1. Each probability $P(x)$ must be between 0 and 1.
2. The total probability must sum to 1 (i.e., $ ext{Sum of } P(x) = 1$).

Graph discrete probability distributions by representing each probability as a bar on a histogram or a similar graphical representation.

The mean (expected value) of a discrete random variable is calculated using the formula:
ext{Mean}(ar{X}) = ext{E}(X) = ext{Sum}(x imes P(x))
Interpretation: When an experiment is repeated many times, the mean value approaches $ar{X}$. Let $x1, x2,
ightarrow x_n$ be the results from each trial.
Example: If a basketball player shoots three free throws 100 times, the mean number may approach a theoretical mean calculated previously.

The mean of a random variable is often referred to as the expected value, denoted as $E(X)$.
Example 1: An 18-year-old male buys a $250,000 life insurance policy. The probability he survives the year is $0.999$, making $P(dies) = 0.001$. The expected value would be calculated considering both the premium collected and the potential payout.
Interpretation: The insurance company can expect a profit of $100 per 18-year-old male client insured.

Standard deviation ($sX$) of a discrete random variable is calculated using the formula: sX = ext{sqrt}igg( ext{Sum}((x - ar{X})^2 imes P(x)) igg)
Alternatively, can use:
s_X = ext{sqrt}igg( ext{Sum}(x^2 imes P(x)) - ar{X}^2 igg)
Example: To find standard deviation given previously defined distributions.

A binomial probability distribution is applicable to experiments with two mutually exclusive outcomes (success and failure).

Example 1: A player with an 80% free throw average shooting three free throws constitutes a binomial experiment ($n = 3$, $p = 0.8$).

Probability of obtaining $x$ successes in $n$ trials is given by:
P(X=x) = inom{n}{x} p^x (1-p)^{n-x}

Graphs for varying $p$ and $n$ result in distinct shapes.
- Higher $p$ shifts the distribution towards success; larger $n$ can lead to a bell shape, indicating increasing trials approximate normal distributions.

A Poisson process follows the occurrence of events with defined average rates under specific conditions:
1. The probability of more than one event in a small series is negligible.
2. Events are independent across different intervals.
Example 1: Cars arriving at a drive-through at an average rate of 2 per minute.

Probability for a Poisson distributed random variable $X$ is given by:
P(X=x) = rac{e^{- ext{λ}} ext{λ}^x}{x!} where $ ext{λ}$ is the average occurrences in the interval.