Discrete and Continuous Probability Distributions: Binomial and 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 = the number of heads in 3 coin flips; possible values: x = 0, 1, 2, or 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, or 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:

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

• $x_i$ = possible value of the random variable

• $P(X = x_i)$ = probability that the random variable equals $x_i$

• $n$ = number of possible values

• Example: Rolling a die: $E(X) = (1)(1/6) + (2)(1/6) + ... + (6)(1/6) = 3.5$

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:

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

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:

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

• $\binom{n}{x}$ = number of ways to choose $x$ successes from $n$ trials

• $p^x$ = probability of $x$ successes

• $(1-p)^{n-x}$ = probability of $n-x$ failures

• Example: In 5 coin flips, what is the probability of exactly 3 heads?

Mean of Binomial Distribution

The mean (expected value) of a binomial distribution is:

$E(X) = n p$

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: Out 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)

The probability of getting exactly $x$ successes in a hypergeometric setting is:

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

• $\binom{K}{x}$ = 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

The mean (expected value) of a hypergeometric distribution is:

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

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.

• Examples: Heights, weights, time intervals.

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.

Comparison Table: Discrete vs. Continuous Random Variables

Feature	Discrete Random Variable	Continuous Random Variable
Possible Values	Specific, separate values (e.g., 0, 1, 2, 3)	Any value within an interval (e.g., 2.5, 2.59, 2.595)
Probability Representation	Probability for each value listed in a table	Probability for intervals, represented by area under curve
Sum of Probabilities	Sum of all probabilities = 1	Total area under curve = 1
Probability of Exact Value	Can be nonzero	Always zero