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Discrete and Continuous Random Variables: Probability Distributions and Functions

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Random Variables

Definition and Types

A random variable is a function that assigns a numerical value to each outcome of a random experiment. Random variables are denoted by uppercase letters (e.g., X), and their possible values are denoted by lowercase letters (e.g., x).

  • Discrete Random Variable: Takes on a finite or countable number of values, usually integers.

  • Continuous Random Variable: Takes on values on a continuous scale, such as heights or weights.

Discrete Random Variables

Probability Mass Function (PMF)

The probability mass function (PMF) of a discrete random variable X is a function p(x) that gives the probability that X takes the value x.

  • Properties:

  • Representation: PMF can be visualized as a bar chart, where the height of each bar represents the probability of each value.

Examples of PMF

  • Coin Toss: Tossing a coin 3 times, X = number of heads. Possible values: 0, 1, 2, 3.

  • Socks Example: Drawing socks from a drawer, X = number of brown socks selected.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable X is .

  • Properties:

    • is non-decreasing

    • as

  • Relationship: The PMF can be obtained from the CDF by finding the size of the jumps at each mass point.

Common Discrete Distributions

Bernoulli Distribution

A Bernoulli random variable X takes values 0 or 1, representing failure or success.

  • PMF:

Binomial Distribution

Represents the number of successes in n independent Bernoulli trials.

  • PMF:

  • Properties:

    • n independent and identical trials

    • Each trial has probability p of success

Geometric Distribution

Number of trials until the first success in repeated Bernoulli trials.

  • PMF: for

Negative Binomial Distribution

Number of trials until r successes are obtained.

  • PMF: for

Hypergeometric Distribution

Sampling without replacement from a finite population.

  • PMF:

  • N: population size, K: number of successes in population, n: sample size, x: number of successes in sample

Poisson Distribution

Models the number of events occurring in a fixed interval of time or space.

  • PMF: for

  • Parameter is the average rate of occurrence.

  • Poisson can be derived as a limit of the binomial distribution as n → ∞ and p → 0 with np = λ.

Continuous Random Variables

Probability Density Function (PDF)

The probability density function (PDF) describes the likelihood of a continuous random variable taking on a value near x.

  • Properties:

  • Probability for an interval:

  • Probability at a single point is zero:

Cumulative Distribution Function (CDF)

The CDF for a continuous random variable is .

  • Relationship:

  • Probability for an interval:

Common Continuous Distributions

Uniform Distribution

All values in an interval are equally likely.

  • PDF on [a, b]:

  • CDF: for

Exponential Distribution

Models waiting times between events.

  • PDF:

  • CDF: for

  • Memoryless property:

Gamma Distribution

Generalizes the exponential distribution; used for modeling waiting times for multiple events.

  • PDF: for

  • Gamma function:

  • If , reduces to exponential distribution.

Normal Distribution

Most common continuous distribution; models many natural phenomena.

  • PDF:

  • Parameters: (mean), (standard deviation)

  • Standard normal: ,

Beta Distribution

Useful for modeling random variables restricted to [0, 1].

  • PDF: for

  • If , reduces to uniform distribution.

Functions of Random Variables

Transformation of Variables

Given a random variable X with density , and Y = g(X), the density function of Y can be found using transformation techniques.

  • If g is monotonic and differentiable:

  • For linear transformations of normal variables: If and , then

Summary Table: Common Discrete and Continuous Distributions

Distribution

Type

Parameters

PMF/PDF

Support

Bernoulli

Discrete

p

0, 1

Binomial

Discrete

n, p

0, 1, ..., n

Geometric

Discrete

p

1, 2, ...

Negative Binomial

Discrete

r, p

x ≥ r

Hypergeometric

Discrete

N, K, n

max(0, n+K-N) ≤ x ≤ min(n, K)

Poisson

Discrete

λ

0, 1, 2, ...

Uniform

Continuous

a, b

a ≤ x ≤ b

Exponential

Continuous

λ

x ≥ 0

Gamma

Continuous

α, λ

x ≥ 0

Normal

Continuous

μ, σ

-∞ < x < ∞

Beta

Continuous

α, β

0 < x < 1

Quantiles

Definition

The p-th quantile of a distribution function F is the value q such that .

  • First quartile:

  • Median:

  • Third quartile:

Key Properties and Applications

  • Discrete random variables are used to model countable outcomes (e.g., number of heads in coin tosses).

  • Continuous random variables are used to model measurements (e.g., time, weight).

  • Probability distributions provide the foundation for statistical inference, hypothesis testing, and modeling real-world phenomena.

Example: Calculating the probability of getting 5 heads in 12 coin tosses using the binomial distribution:

Example: Probability that a light bulb lasts at least 5 more hours, given it has already lasted 3 hours, using the exponential distribution:

Example: Probability that none of 4 polluting trucks are sampled from 24 trucks, using the hypergeometric distribution:

Additional info: The notes cover foundational concepts for probability distributions, including both discrete and continuous cases, and provide formulas, properties, and examples for each major distribution relevant to college statistics.

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