Probability Distributions

In earlier topics, we calculated probabilities for individual events from scratch. However, mathematicians noticed that certain types of random experiments occur so frequently in the real world that they follow predictable, standardized mathematical patterns.

Instead of deriving the probability from scratch every time, we can use these standardized models, known as Theoretical Probability Distributions. In this section, we will cover the most common discrete probability distributions: the Bernoulli, Binomial, and Poisson distributions.

1. Bernoulli Distribution (The Foundation)

The simplest of all probability distributions is the Bernoulli distribution. It models a single trial of a random experiment that has exactly two possible outcomes.

• Success (S): Assigned the numerical value $X = 1$. It occurs with probability $p$.
• Failure (F): Assigned the numerical value $X = 0$. It occurs with probability $q = 1 - p$.

Any experiment that fits this criteria is called a Bernoulli Trial. * Examples: Flipping a coin (Heads = Success, Tails = Failure), testing a lightbulb (Works = Success, Defective = Failure), passing or failing an exam.

Properties of a Bernoulli RV: * Expected Value: $E(X) = p$ * Variance: $Var(X) = p \cdot q$

2. Binomial Distribution

What happens if you perform a Bernoulli trial not just once, but multiple times in a row? This brings us to the Binomial Distribution.

The Binomial Distribution models the number of successes ($x$) in a fixed number of independent Bernoulli trials ($n$).

Conditions for a Binomial Experiment:

1. The experiment consists of a fixed number of trials, $n$.
2. Each trial has only two possible outcomes (Success or Failure).
3. The probability of success, $p$, remains constant from trial to trial.
4. The trials are completely independent of each other (one trial doesn't affect the next).

The Binomial Formula

If a random variable $X$ follows a binomial distribution, the probability of getting exactly $x$ successes in $n$ trials is given by:

$$ P(X = x) = \binom{n}{x} p^x q^{n-x} $$

• $n$: Total number of trials.
• $x$: Number of successes desired ($x = 0, 1, 2, \dots, n$).
• $p$: Probability of success on a single trial.
• $q$: Probability of failure on a single trial ($1 - p$).
• $\binom{n}{x}$: "n choose x" (combinations), calculated as $\frac{n!}{x!(n-x)!}$. This accounts for the different orders in which the successes and failures can occur.

Properties of a Binomial RV:

• Mean (Expected Value): $\mu = n \cdot p$
• Variance: $\sigma^2 = n \cdot p \cdot q$
• Standard Deviation: $\sigma = \sqrt{n \cdot p \cdot q}$

Example: A biased coin has a 60% chance of landing on Heads. If you flip it 5 times, what is the probability of getting exactly 3 Heads? * $n = 5$ * $x = 3$ * $p = 0.6$ * $q = 0.4$ * $P(X = 3) = \binom{5}{3} (0.6)^3 (0.4)^2 = 10 \times (0.216) \times (0.16) = 0.3456$ (or 34.56%).

3. Poisson Distribution

While the Binomial distribution models the number of successes in a fixed number of trials, the Poisson Distribution models the number of successes (or events) occurring in a fixed interval of time or space.

The Poisson distribution is often used to model rare events or rates of occurrence where $n$ is theoretically infinite and $p$ is very small.

• Examples: The number of emails arriving in your inbox per hour, the number of potholes on a 10-mile stretch of road, the number of radioactive particles emitted per second.

Conditions for a Poisson Experiment:

1. Events occur independently of one another.
2. The average rate (occurrences per interval) is constant.
3. Two events cannot occur at the exact same infinitesimal instant in time/space.

The Poisson Formula

If a random variable $X$ follows a Poisson distribution with an average rate of $\lambda$ (lambda) occurrences per interval, the probability of exactly $x$ occurrences is:

$$ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} $$

• $\lambda$: The average (expected) number of occurrences in the interval.
• $x$: The actual number of occurrences we want the probability for ($x = 0, 1, 2, \dots$).
• $e$: Euler's number, approximately 2.71828...

Properties of a Poisson RV:

A unique and defining mathematical characteristic of the Poisson distribution is that its mean and variance are equal! * Mean (Expected Value): $\mu = \lambda$ * Variance: $\sigma^2 = \lambda$ * Standard Deviation: $\sigma = \sqrt{\lambda}$

Example: A call center receives an average of 4 calls per minute. What is the probability of receiving exactly 2 calls in a randomly selected minute? * $\lambda = 4$ * $x = 2$ * $P(X = 2) = \frac{e^{-4} 4^2}{2!} = \frac{0.0183 \times 16}{2} = 0.1465$ (or 14.65%).

4. Normal Distribution (Brief Overview)

While the distributions above are discrete, the Normal Distribution (the famous "bell curve") is the most important continuous probability distribution.

It is symmetric, centered around its mean ($\mu$), and its spread is determined by its standard deviation ($\sigma$). Due to the Central Limit Theorem, many natural phenomena (heights, test scores, measurement errors) naturally follow a normal distribution, making it the cornerstone of modern statistics.