Module 5: Discrete Probability Distributions - Cumulative Probabilities and the Poisson Distribution

Definition and Purpose: Historically, calculations have focused on exact probabilities (e.g., $P(X = 3)$ ). Cumulative probabilities expand this by calculating the likelihood of a range of events (e.g., $P(X \le 2)$ , P(X < 2), or $P(X \ge 3)$ ).
Distinction between Exact and Cumulative:
- Exact Probability: Indicated by the equal sign ( $=$ ), representing a specific point in the distribution.
- Cumulative Probability: Indicated by inequality signs (\le, <, \ge, >), representing an accumulation of points.
Binomial Distribution Recap:
- Involves $n$ independent trials.
- The probability of a successful event from trial to trial remains constant, denoted as $\pi$ .

Scenario Parameters:
- Probability of success: $\pi = 0.1$
- Number of trials: $n = 4$
- Possible outcomes: $\{0, 1, 2, 3, 4\}$
Objective: Calculate the probability that the number of successes is less than or equal to two ( $P(X \le 2)$ ).
Mathematical Representation: The cumulative probability is the sum of the individual probabilities for the relevant events:
- $P(X \le 2) = P(X = 2) + P(X = 1) + P(X = 0)$
Calculating Individual Probabilities: Using the binomial formula $P(X = x) = \binom{n}{x} \pi^x (1-\pi)^{n-x}$ , where $\binom{n}{x}$ represents "n choose x":
- For $x = 2$ : $P(X = 2) = \binom{4}{2} 0.1^2 \times 0.9^{4-2}$
- For $x = 1$ : $P(X = 1) = \binom{4}{1} 0.1^1 \times 0.9^{4-1}$
- For $x = 0$ : $P(X = 0) = \binom{4}{0} 0.1^0 \times 0.9^{4-0}$
Calculated Result: Summing these values yields a final probability of $0.9963$ .

Exact Probabilities: The function BINOM.DIST(number_s, trials, probability_s, cumulative) is used.
- Setting cumulative to FALSE (or $0$ ) provides the exact probability for that specific value.
Cumulative Probabilities: Setting cumulative to TRUE (or $1$ ) instructs Excel to accumulate every probability from the specified value down to zero.
- Example: BINOM.DIST(2, 4, 0.1, TRUE) directly equals $0.9963$ .

The Complement Rule: Since all probabilities in a sample space must add up to $1$ , the probability of an event can be found by subtracting the probability of its complement from $1$ .
- Rule: P(X \le 2) = 1 - P(X > 2)
- Expansion: $1 - (P(X = 3) + P(X = 4))$
Inclusion vs. Exclusion:
- If a value is inclusive on one side of the calculation, it must be exclusive on the other side to avoid double-counting.
- In discrete cases, a strict inequality can be rewritten as an inclusive one: P(X < 2) is identical to $P(X \le 1)$ . This represents the sum of $P(X = 0) + P(X = 1)$ .
Discrete vs. Continuous Distinction: These rules for shifting inequalities (e.g., P(X < 2) = P(X \le 1)) apply strictly to the discrete data discussed in Chapter 5. They do not apply to the continuous data covered in Chapter 6.

Context: This technique relates to the use of Binomial Tables, historical artifacts used for precomputing probabilities before high-powered computing was common.
The Subtraction Rule: To find the exact probability of $k$ using only cumulative data, subtract the cumulative probability of $k-1$ from the cumulative probability of $k$ .
- Formula: $P(X = k) = P(X \le k) - P(X \le k-1)$
- Example: To find $P(X = 3)$ , take the probability of all outcomes up to three ( $P(X \le 3)$ ) and subtract the outcomes that are strictly less than three ( $P(X \le 2)$ ). This leaves only the probability of exactly three.

Comparison to Binomial Distribution:
- Binomial: discrete events within a discrete number of trials ( $n$ ).
- Poisson: discrete events measured over a continuous "area of opportunity."
Area of Opportunity: This is a continuous interval in which events can occur, defined by:
1. Time: e.g., people walking into a store per hour.
2. Surface Area: e.g., mosquito bites on a patch of skin.
3. Length: e.g., defects in a wire.
Examples of Poisson Scenarios:
- Customer Traffic: If the average rate ( $\lambda$ ) is two people per hour, the Poisson distribution can predict the probability of exactly three people entering in the next hour.
- Entomology: If the average rate is $1.5$ mosquito bites per skin area, one can calculate the probability of three bites occurring on a specific patch.

Constant Rate: The probability that an event occurs in a given area of opportunity remains the same across all similar areas.
- Real-world Caveat: Rates often fluctuate (e.g., store traffic is higher during lunch hours), so researchers must standardize periods to maintain Poisson applicability.
Independence: The number of events occurring in one area of opportunity is independent of the number of events in any other area. An event in hour one does not influence the likelihood of an event in hour two.
Shrinking Area Rule: As the area of opportunity becomes smaller, the probability and likelihood of two or more events occurring simultaneously approaches zero ( $0$ ).
- Example: The chance of two mosquito bites occurring in the exact same spot is much lower than them occurring in a large area of skin.
Rate Parameter ( $\lambda$ ): The Greek letter lambda ( $\lambda$ ) represents the average number of events per unit area of opportunity. This is the central parameter of the Poisson formula.

Student Question: "Which ones am I going to color in? If it's strictly less than two, which ones do I color in?"
Response: For the discrete case, if the criteria is strictly less than two (P(X < 2)), you color in (or include) zero ( $0$ ) and one ( $1$ ). You stop there because two is not included.
Student Question: Regarding the visual representation of P(X \ge 2) = 1 - P(X < 2).
Response: If you want the sum of probabilities for two, three, four, and five, it is faster to take the total sum of all probabilities ( $1$ ) and subtract the ones you don't want (zero and one). This is a shortcut for calculations.
Note on Statistics A1 Test:
- The test is "take-home" and "open-book."
- It covers the first four chapters (basic theory and probability).
- Excel skills are required; students refer to TUT 2 (Tutorial 2) recordings for practice.