Binomial and Poisson Distributions: Study Notes for Introductory Statistics

Binomial Distribution

Definition and Properties

The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.

• Outcomes categorized in two ways: Each trial results in either a success (S) or failure (F).

• Independence: Each trial is independent of the others.

• Fixed number of trials: The total number of trials, denoted as n, is predetermined.

• Probability of success: The probability of success in each trial is p, and the probability of failure is q = 1 - p.

Binomial Probability Formula

The probability of observing exactly x successes in n trials is given by:

$P(x) = \frac{n!}{x!(n-x)!} p^x q^{n-x}$

• n: Number of trials

• x: Number of successes

• p: Probability of success

• q: Probability of failure ($q = 1 - p$)

Parameters of the Binomial Distribution

• Mean (Expected Value): $M = n p$

• Standard Deviation: $\sigma = \sqrt{n p q}$

Example: If you flip a coin 10 times (n=10), and the probability of heads (success) is 0.5, the mean number of heads is $M = 10 \times 0.5 = 5$.

Poisson Distribution

Definition and Properties

The Poisson distribution is a discrete probability distribution that models the number of occurrences of an event in a fixed interval of time or space, given that these events occur independently and at a constant average rate.

• Number of occurrences: Counts the number of times an event occurs in a given interval.

• Independence: Occurrences are independent.

• Uniform distribution: The probability of occurrence is uniform throughout the interval.

• Randomness: Events occur randomly.

Poisson Probability Formula

The probability of observing exactly x occurrences in an interval is:

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

• M: Mean number of occurrences in the interval

• x: Number of occurrences

• e: Euler's number (approximately 2.718)

Parameters of the Poisson Distribution

• Mean: $M$

• Standard Deviation: $\sigma = \sqrt{M}$

Example: If the average number of emails received per hour is 3, the probability of receiving exactly 2 emails in an hour is $P(2) = \frac{3^2 e^{-3}}{2!}$.

Relationship Between Binomial and Poisson Distributions

Poisson Approximation to Binomial

The Poisson distribution can approximate the binomial distribution when the number of trials is large and the probability of success is small. Specifically, if n > 100 or np < 10, then:

$P_{bin}(x) \approx P_{pois}(x)$ where $M = n p$

• Use Poisson approximation: When binomial calculations are cumbersome due to large n and small p.

Sample Space and Counting Principles

Sample Space

The sample space is the set of all possible outcomes of an experiment.

• Example: For tossing a coin three times, the sample space is {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}.

Counting Trials and Calculating Probabilities

Use factorial notation to count the number of ways events can occur.

• Permutation formula: $P(n, r) = \frac{n!}{(n-r)!}$

• Example: $P(7, 3) = \frac{7!}{4!} = 210$

Identifying Binomial and Poisson Scenarios

Which Distribution to Use?

• Binomial: Use when there are a fixed number of independent trials, each with two possible outcomes.

• Poisson: Use when counting occurrences over a continuous interval, with events happening independently and at a constant rate.

Example: If you are counting the number of defective items in a batch (fixed number), use binomial. If you are counting the number of phone calls received in an hour, use Poisson.

Summary Table: Binomial vs. Poisson Distribution

Property	Binomial Distribution	Poisson Distribution
Type of Experiment	Fixed number of trials	Occurrences in interval
Outcomes	Success/Failure	Number of events
Parameters	n (trials), p (probability)	M (mean rate)
Mean	$n p$	$M$
Standard Deviation	$\sqrt{n p q}$	$\sqrt{M}$
Formula	$P(x) = \frac{n!}{x!(n-x)!} p^x q^{n-x}$	$P(x) = \frac{M^x e^{-M}}{x!}$

Additional info: The notes emphasize the importance of understanding when to use each distribution and how to approximate the binomial with the Poisson under certain conditions. Practice problems include listing sample spaces, counting trials, and identifying which distribution applies to a scenario.