Discrete and Continuous Probability Distributions: Binomial & Hypergeometric

Discrete Probability Distributions

Random Variables

A random variable is a numerical value that results from a random event or experiment. Random variables are typically denoted by capital letters (e.g., X, Y), while their possible values are represented by lowercase letters (e.g., x, y).

• Definition: A variable whose value is determined by the outcome of a random process.

• Example: X = number of heads in 3 coin flips; possible values: x = 0, 1, 2, 3.

Probability Distribution of a Random Variable

The probability distribution of a random variable lists all possible values the variable can take and the probability of each value occurring.

• Notation: Capital letter (X) for the random variable before the experiment; lowercase letter (x) for the actual value after the experiment.

• Example: Rolling a die: X = outcome, x = 1, 2, 3, 4, 5, 6.

Discrete Random Variables

A discrete random variable can only take specific, separate values (such as whole numbers), and nothing in between. Each value has a probability between 0 and 1, and the sum of all probabilities is 1.

• Example: Rolling a die; possible outcomes are 1, 2, 3, 4, 5, 6.

Expected Value (Mean) of a Discrete Random Variable

The expected value (mean) is the average value you expect if the random experiment is repeated many times.

• Formula:

$E(X) = \mu = \sum_{i=1}^{n} x_i \cdot P(X = x_i)$

• Example: Rolling a die:

  • Values: 1, 2, 3, 4, 5, 6

  • Probabilities: 1/6 each

  • $E(X) = (1)(1/6) + (2)(1/6) + ... + (6)(1/6) = 3.5$

Variance of a Discrete Random Variable

Variance measures how spread out the values are from the mean.

• If variance is small, values are usually close to the mean.

• If variance is large, values are usually more spread out.

• Formula:

$Var(X) = \sigma^2 = E[(X - \mu)^2] = \sum_{i=1}^{n} (x_i - \mu)^2 \cdot P(X = x_i)$

Binomial Distribution

Definition and Properties

The binomial distribution is used when each trial has two possible outcomes: success or failure. It models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Fixed number of trials ($n$)

• Each trial has two outcomes (success or failure)

• Probability of success ($p$) is the same for each trial

• Trials are independent

Probability Mass Function (PMF)

The probability mass function gives the probability of exactly $x$ successes in $n$ trials.

$P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}$

• $p$ = probability of success on a single trial

• $n$ = number of trials

• $x$ = number of successes

• $1-p$ = probability of failure

Mean of Binomial Distribution

• Formula:

$E(X) = np$

Hypergeometric Distribution

Definition and Properties

The hypergeometric distribution calculates the probability of getting $x$ successes when you randomly pick a sample from a finite group without replacement. The probability changes each time because items are not replaced.

• $N$ = population size

• $K$ = number of items of interest (successes in population)

• $n$ = number of trials (sample size)

• $x$ = number of successes drawn

Example: From a team of 50 employees ($N=50$), 10 are from finance ($K=10$). If 8 are selected ($n=8$), what is the probability that exactly 2 are from finance?

Probability Mass Function (PMF)

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

• $\binom{K}{x}$ = ways to choose $x$ successes from $K$

• $\binom{N-K}{n-x}$ = ways to choose $n-x$ failures from $N-K$

• $\binom{N}{n}$ = total ways to choose $n$ items from $N$

Mean of Hypergeometric Distribution

• Formula:

$E(X) = n \frac{K}{N}$

Continuous Probability Distributions

Definition and Properties

A continuous probability distribution describes variables that can take infinitely many values in a range. Instead of tables, functions are used to describe probabilities.

• A continuous variable can take any value within an interval (e.g., time could be 2.5 hours, 2.59 hours, etc.).

• Unlike discrete random variables, you cannot list all possible values in a table because there are infinitely many.

Continuous Random Variable

Probabilities for continuous random variables come from areas under the curve over intervals, not single points.

• The area under the curve = probability that the variable takes a value in that interval.

• The probability of one exact value is zero.

Examples of Continuous Distributions

• Normal distribution

• Uniform distribution

• Exponential distribution

*Additional info: Expanded definitions, formulas, and examples were added for clarity and completeness.*