Discrete Probability Distributions: Binomial, Poisson, and Hypergeometric Distributions

Discrete Probability Distributions

Introduction to Random Variables & Probability Distributions

A random variable is a numerical value determined by the outcome of a random experiment. Random variables can be classified as either discrete or continuous:

• Discrete Random Variable (DRV): Takes on countable values (e.g., number of defective bulbs in a batch).

• Continuous Random Variable (CRV): Can take on any value within a range (e.g., height, time).

A probability distribution lists the probabilities associated with each possible value of a random variable.

Example: Probability Distribution Table

Suppose a lottery has the following profit outcomes:

Profit	Probability
-$1.00	0.40
$0.00	0.35
$5.00	?
$1,000,000.00	0.01

To find the missing probability, recall that the sum of all probabilities must equal 1.

Criteria for a Probability Distribution:

• All probabilities are between 0 and 1.

• The sum of all probabilities is 1.

Identifying Discrete Random Variables

Discrete random variables are those that can be counted. For example, the number of defective lightbulbs in a batch is discrete, while the time to complete a race is continuous.

Calculating Probabilities from Distributions

To find the probability of a certain event (e.g., at most 2 sodas per day), sum the probabilities for all relevant outcomes.

Mean (Expected Value), Variance, and Standard Deviation of Discrete Random Variables

Expected Value (Mean)

The expected value (mean) of a discrete random variable is calculated by multiplying each value by its probability and summing the results:

$\mu = \sum [x \cdot P(x)]$

Example: For the number of kids per household:

# of Kids	Probability
0	0.15
1	0.60
2	0.25

Expected value: $0 \times 0.15 + 1 \times 0.60 + 2 \times 0.25 = 1.10$

Variance and Standard Deviation

The variance and standard deviation measure the spread of a probability distribution:

$\sigma^2 = \sum [(x - \mu)^2 \cdot P(x)]$

$\sigma = \sqrt{\sigma^2}$

Construct a table with columns for $x$, $P(x)$, $(x-\mu)^2$, and $(x-\mu)^2 \cdot P(x)$ to compute these values.

Binomial Distribution

The Binomial Experiment

A binomial experiment consists of a fixed number of independent trials, each with two possible outcomes: success or failure. The probability of success ($p$) is constant for each trial.

• Number of trials: $n$

• Probability of success: $p$

• Probability of failure: $q = 1 - p$

To be a binomial experiment, the following must be true:

• Fixed number of trials

• Each trial is independent

• Each trial has only two outcomes

• Probability of success is the same for each trial

Binomial Probability Formula

The probability of getting exactly $k$ successes in $n$ trials is given by:

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

where $\binom{n}{k}$ is the number of combinations of $n$ items taken $k$ at a time.

Mean and Standard Deviation of Binomial Distribution

For a binomial distribution:

$\mu = n p$

$\sigma = \sqrt{n p q}$

Using Technology (TI-84) for Binomial Probabilities

To find binomial probabilities:

• Use binompdf for exact probabilities

• Use binomcdf for cumulative probabilities (e.g., "at most", "at least")

Poisson Distribution

Introduction to Poisson Distribution

The Poisson distribution models the number of occurrences of an event in a fixed interval of time or space, given the average rate ($\lambda$) of occurrence.

• Mean: $\lambda$

• Variance: $\lambda$

The probability of observing exactly $x$ events is:

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

When to Use Poisson vs. Binomial

• Binomial: Fixed number of trials, each with two outcomes, constant probability of success.

• Poisson: Counts of events in a fixed interval, events occur independently, mean rate is known.

Using Technology (TI-84) for Poisson Probabilities

To find Poisson probabilities:

• Use poissonpdf for exact probabilities

• Use poissoncdf for cumulative probabilities

Poisson Approximation to Binomial

When $n$ is large and $p$ is small, the binomial distribution can be approximated by the Poisson distribution with $\lambda = n p$.

Hypergeometric Distribution

Introduction to Hypergeometric Distribution

The hypergeometric distribution models the probability of $x$ successes in $n$ draws from a finite population of size $N$ containing $r$ successes, without replacement.

The probability is given by:

$P(x) = \frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}}$

Key Differences from Binomial:

• Trials are not independent (no replacement)

• Probability of success changes on each draw

Choosing the Correct Distribution

• Binomial: Sampling with replacement or from a very large population.

• Hypergeometric: Sampling without replacement from a finite population.

Summary Table: Key Probability Distributions

Distribution	Scenario	Parameters	Formula
Binomial	Fixed # of independent trials, 2 outcomes	n, p	$P(X = k) = \binom{n}{k} p^k q^{n-k}$
Poisson	Count of events in interval, mean rate known	$\lambda$	$P(x) = \frac{\lambda^x e^{-\lambda}}{x!}$
Hypergeometric	Draws without replacement from finite group	N, r, n	$P(x) = \frac{\binom{r}{x} \binom{N-r}{n-x}}{\binom{N}{n}}$

Additional info: This guide covers the core concepts, formulas, and distinctions among the main discrete probability distributions encountered in introductory statistics. Practice problems and calculator instructions are included to reinforce understanding and application.