Discrete Probability Distributions: Study Notes

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 items in a batch, dice rolls).

• Continuous Random Variable (CRV): Takes 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: The number of sodas consumed per day or the number of prizes won in a raffle are examples of discrete random variables.

Probability Distribution Tables

Probability distributions are often represented in tables, showing each value of the random variable and its probability. The sum of all probabilities must equal 1, and each probability must be between 0 and 1.

Value (X)	Probability P(X)
0	0.10
1	0.20
2	0.40
3	0.20
4	0.10

Criteria for a Probability Distribution:

• 0 ≤ P(X) ≤ 1 for every X

• Sum of all P(X) = 1

Mean (Expected Value) of a Discrete Random Variable

The mean or expected value 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 (X)	Probability P(X)
0	0.15
1	0.60
2	0.25

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

Variance and Standard Deviation of Discrete Random Variables

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

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

• Standard Deviation: $\sigma = \sqrt{\sigma^2}$

Example: For the number of complaints received daily:

# of Complaints (X)	Probability P(X)
0	0.45
1	0.30
2	0.20
3	0.05

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.

• n: Number of trials

• p: Probability of success

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

Example: Flipping a coin multiple times or drawing marbles with replacement.

Binomial Probability Formula

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

$P(X = x) = \left( \begin{array}{c} n \\ x \end{array} \right) p^x q^{n-x}$

where $\left( \begin{array}{c} n \\ x \end{array} \right)$ is the binomial coefficient (number of ways to choose x successes from n trials).

Mean and Standard Deviation of Binomial Distribution

For a binomial distribution:

• Mean: $\mu = np$

• Variance: $\sigma^2 = npq$

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

Using Technology (TI-84) for Binomial Probabilities

To find binomial probabilities:

• For exact probabilities, use binompdf

• For cumulative probabilities, use binomcdf

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 (λ) of occurrence.

• Mean: $\mu = \lambda$

• Variance: $\sigma^2 = \lambda$

The probability of observing exactly x events is:

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

Using Technology (TI-84) for Poisson Probabilities

To find Poisson probabilities:

• For exact probabilities, use poissonpdf

• For cumulative probabilities, use poissoncdf

Hypergeometric Distribution

Introduction to Hypergeometric Distribution

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

• Population size: N

• Number of successes in population: r

• Sample size: n

• Number of observed successes: x

The probability is given by:

$P(X = x) = \frac{\left( \begin{array}{c} r \\ x \end{array} \right) \left( \begin{array}{c} N - r \\ n - x \end{array} \right)}{\left( \begin{array}{c} N \\ n \end{array} \right)}$

Example: Drawing marbles from a bag without replacement, or selecting defective items from a shipment.

Choosing the Appropriate Distribution

• Binomial: Fixed number of independent trials, constant probability, with replacement.

• Hypergeometric: Fixed number of dependent trials, probability changes, without replacement.

• Poisson: Counts of events in a fixed interval, given average rate.

Additional info: These distributions are foundational for inferential statistics, hypothesis testing, and real-world applications such as quality control, survey analysis, and reliability engineering.