Cumulative distribution functions (CDFs) are a key tool for understanding discrete random variables. They give us the probability that a variable is less than or equal to a specific value, making it easier to calculate probabilities for ranges and compare distributions.
CDFs build on probability mass functions (PMFs) by summing up probabilities for all values up to a given point. This cumulative approach helps us visualize the overall shape of a distribution and quickly find probabilities for different scenarios, connecting to broader concepts in probability theory.
Cumulative Distribution Functions for Discrete Variables
Definition and Properties
- Cumulative distribution function (CDF) for discrete random variable X gives probability X is less than or equal to x for all real numbers x
- Defined as F(x) = P(X ≤ x) for all x in real number line
- Non-decreasing function F(x1) ≤ F(x2) for all x1 < x2
- Range between 0 and 1 inclusive 0 ≤ F(x) ≤ 1 for all x
- Step function for discrete random variables with jumps at possible values of X
- Approaches 0 as x approaches negative infinity and 1 as x approaches positive infinity
- Right-continuous includes endpoint at each jump
Examples and Visualizations
- For a six-sided die, CDF would have jumps at x = 1, 2, 3, 4, 5, 6
- F(x) = 0 for x < 1, F(x) = 1/6 for 1 ≤ x < 2, F(x) = 2/6 for 2 ≤ x < 3, etc.
- Visualize as staircase function with steps at each possible value
- For Poisson distribution with λ = 3, CDF would have jumps at x = 0, 1, 2, 3, ...
- F(x) = e^(-3) for 0 ≤ x < 1, F(x) = e^(-3) * (1 + 3) for 1 ≤ x < 2, etc.
Probabilities with CDFs
Calculating Specific Probabilities
- Probability X exactly equals value k calculated as P(X = k) = F(k) - F(k^-)
- F(k^-) represents left-hand limit of F at k
- Probability X less than value k given by P(X < k) = F(k^-)
- Probability X greater than value k calculated as P(X > k) = 1 - F(k)
- Probability X between two values a and b (inclusive) given by P(a ≤ X ≤ b) = F(b) - F(a^-)
- For open intervals adjust calculation P(a < X < b) = F(b^-) - F(a)
Application Examples
- Six-sided die probability of rolling 4 or less P(X ≤ 4) = F(4) = 4/6
- Probability of waiting more than 2 minutes in Poisson process (λ = 0.5 per minute) P(X > 2) = 1 - F(2)
- Binomial distribution (n=10, p=0.3) probability of 3 to 7 successes P(3 ≤ X ≤ 7) = F(7) - F(2)
- Geometric distribution (p=0.2) probability of success on 3rd to 5th trial P(3 ≤ X ≤ 5) = F(5) - F(2)
Constructing CDFs from PMFs
Step-by-Step Process
- Identify all possible values of random variable X
- Calculate cumulative probability F(x_i) by summing probabilities of all values less than or equal to x_i
- F(x_i) = Σ P(X = x_j) for all x_j ≤ x_i
- Plot cumulative probabilities as step function with jumps at each possible value of X
- Ensure CDF right-continuous by including endpoint at each jump
- Extend CDF to all real numbers F(x) = 0 for x less than smallest possible value of X
- F(x) = 1 for x greater than or equal to largest possible value of X
- Verify resulting CDF satisfies all properties of valid cumulative distribution function
Practical Examples
- Construct CDF for rolling a fair six-sided die
- PMF: P(X = k) = 1/6 for k = 1, 2, 3, 4, 5, 6
- CDF: F(x) = 0 for x < 1, F(x) = 1/6 for 1 ≤ x < 2, F(x) = 2/6 for 2 ≤ x < 3, etc.
- Construct CDF for Binomial distribution with n = 3, p = 0.5
- PMF: P(X = 0) = 1/8, P(X = 1) = 3/8, P(X = 2) = 3/8, P(X = 3) = 1/8
- CDF: F(x) = 0 for x < 0, F(x) = 1/8 for 0 ≤ x < 1, F(x) = 4/8 for 1 ≤ x < 2, etc.
CDFs vs PMFs
Key Differences and Relationships
- PMF derived from CDF for discrete random variables by taking difference between consecutive CDF values
- P(X = x_i) = F(x_i) - F(x_i^-)
- CDF cumulative sum of PMF values F(x) = Σ P(X = x_i) for all x_i ≤ x
- PMF represents probability of random variable taking specific values
- CDF represents probability of random variable being less than or equal to given value
- CDF always non-decreasing function PMF can have any non-negative values summing to 1
- CDF continuous from right PMF defined only for discrete values of random variable
- Both provide complete descriptions of probability distribution of discrete random variable
Practical Applications
- Use PMF when interested in exact probabilities of specific outcomes (rolling a 6 on a die)
- Use CDF when interested in cumulative probabilities or ranges (rolling 4 or less on a die)
- PMF useful for calculating expected values and variances
- CDF useful for finding percentiles and quantiles of distribution
- In data analysis PMF often visualized as histogram CDF as step function
- For large datasets CDF smoother and easier to interpret than PMF
- In hypothesis testing CDFs used to compute p-values and critical values