How do you read a discrete CDF chart?

Find the x-value of interest on the horizontal axis and read upward to where it meets the step function. The y-coordinate at that point gives the cumulative probability P(X ≤ x). The step function has horizontal segments between possible values and vertical jumps at each outcome. Each jump height equals the probability mass at that point.

What is the difference between CDF and PMF?

The probability mass function (PMF) gives the probability of exactly one value: P(X = k). The cumulative distribution function (CDF) sums all PMF values up to and including k: F(k) = sum of P(X = i) for all i ≤ k. PMF shows individual probabilities, while CDF shows accumulated probability. You can find PMF values from the CDF by computing differences between consecutive steps.

Why do discrete CDFs have step patterns?

Discrete CDFs form step functions because probability only exists at specific countable values. Between these values, no outcomes are possible, so the CDF remains constant. The function only increases at points where the random variable can take values, creating jumps. Each jump height equals the probability mass at that point, and the horizontal segments show regions with no probability mass.

When should you use CDF instead of PMF?

Use CDF when you need to find probabilities for ranges or cumulative questions like 'at most k successes' or 'no more than x events.' Use PMF when you need the probability of exactly one specific value. CDF is particularly useful for computing interval probabilities using subtraction: P(a < X ≤ b) = F(b) - F(a).

Cumulative Distribution Function(CDF) of Discrete Distributions

Interactive visualization of cumulative distribution functions (CDF)

Discrete Uniform CDF

CDF rises uniformly in equal steps

Minimum Value (a): 1

Maximum Value (b): 6

CDF Explanation

The cumulative distribution function (CDF) for the discrete uniform distribution is $F(k) = \frac{\lfloor k \rfloor - a + 1}{b - a + 1}$ for $a \leq k \leq b$ . The CDF increases in equal steps of $\frac{1}{n}$ where $n = b - a + 1$ is the number of possible values. Each step represents one additional outcome being included in the cumulative probability. The CDF reaches 1.0 at the maximum value $b$ and remains at 1.0 for all larger values.

Selecting a Distribution

Adjusting Distribution Parameters

Reading the CDF Chart

Understanding Step Functions in Discrete CDFs

Finding Specific Cumulative Probabilities

Comparing Distribution Shapes

Interpreting Parameter Effects on CDF Shape

What is a Cumulative Distribution Function?

CDF vs PMF for Discrete Distributions

Related Probability Tools and Concepts

Selecting a Distribution

The visualizer displays six discrete probability distributions across tabs at the top. Click any tab to switch between Discrete Uniform, Binomial, Geometric, Negative Binomial, Hypergeometric, and Poisson distributions. Each distribution models a different probabilistic scenario, from equally likely outcomes to rare event counting. The active tab is highlighted, and the chart immediately updates to show the cumulative distribution function for that distribution with default parameter values.

Adjusting Distribution Parameters

Each distribution has parameter sliders in the controls panel below the distribution name. Drag the sliders to change values:

Discrete Uniform uses minimum value (a) and maximum value (b) sliders to set the range.

Binomial adjusts number of trials (n) and success probability (p).

Geometric controls only success probability (p).

Negative Binomial sets number of successes (r) and success probability (p).

Hypergeometric configures population size (N), success states (K), and number of draws (n).

Poisson adjusts the rate parameter lambda (λ).

The chart updates instantly as you move sliders. Parameter values display next to each slider label.

Reading the CDF Chart

The cumulative distribution function appears as a step chart with discrete jumps. The x-axis shows possible values (k), and the y-axis shows cumulative probability

F(k) = P(X \leq k)

ranging from 0 to 1. Each horizontal segment represents the probability that the random variable is less than or equal to that x-value. Vertical jumps occur at each possible outcome, with jump height equal to

P(X = k)

. The rightmost point always reaches probability 1.0, meaning all outcomes up to that point account for the entire probability mass.

Hover over any point to see exact values. The tooltip displays the k-value and corresponding cumulative probability to six decimal places.

Understanding Step Functions in Discrete CDFs

Discrete CDFs form step functions rather than smooth curves because probability concentrates at specific points. Between integer values, the CDF remains constant—if no outcome can occur at

k = 2.5

, then

F(2.5) = F(2)

. The function only increases at values where outcomes are possible. This creates the characteristic staircase pattern where each step's height equals the probability mass at that point. The step-after line type shows this clearly: the line extends horizontally from each point, then jumps vertically to the next level.

Compare this to continuous distributions, where CDFs rise smoothly without jumps.

Finding Specific Cumulative Probabilities

To find

P(X \leq k)

for any value k, locate k on the x-axis and read upward to the step function. The y-coordinate at that point gives the cumulative probability. For example, if the chart shows 0.842 at

k = 5

, then there's an 84.2% chance the random variable is 5 or less.

You can also calculate probabilities for ranges using the CDF values. To find

P(a < X \leq b)

, subtract:

F(b) - F(a)

. Hover over both endpoints to get their cumulative probabilities, then compute the difference. The visualization makes these probability intervals visually apparent as vertical distances between steps.

Comparing Distribution Shapes

Switch between distribution tabs to compare how different probabilistic mechanisms create different CDF patterns. The Discrete Uniform CDF rises in equal-sized steps. The Binomial CDF typically shows an S-curve shape when p is near 0.5, with steeper increases near the center. The Geometric and Negative Binomial CDFs start low and rise gradually, with the rate depending on success probability. The Hypergeometric CDF resembles binomial but with constraints from finite population sampling. The Poisson CDF rises most rapidly near lambda, with shape determined by the rate parameter.

Experiment with parameters to see how they affect the rate of increase and spread of the CDF.

Interpreting Parameter Effects on CDF Shape

For Binomial, increasing n spreads the CDF over more values, while changing p shifts where the steepest rise occurs—left for small p, right for large p. For Geometric, smaller p values create gentler slopes as more trials are needed on average. The Poisson CDF becomes more spread out as lambda increases, with the steepest rise occurring near the lambda value. Hypergeometric sampling without replacement creates dependencies that compress or expand the CDF compared to binomial sampling with replacement.

Watch how the CDF evolves as you adjust parameters. Steeper rises indicate probability mass concentrated in a narrow range, while gradual rises show probability spread across many values.

What is a Cumulative Distribution Function?

A cumulative distribution function (CDF) gives the probability that a random variable takes a value less than or equal to x:

F(x) = P(X \leq x)

. For discrete distributions, this is computed by summing the probability mass function (PMF) values:

F(k) = \sum_{i \leq k} P(X = i)

. The CDF always starts at 0 and increases to 1 as x increases, never decreasing. It answers questions like "What's the chance of getting 3 or fewer successes?" rather than "What's the chance of exactly 3 successes?"

For comprehensive theory on cumulative distribution functions including mathematical properties and applications, see cumulative distribution function theory page.

CDF vs PMF for Discrete Distributions

The probability mass function (PMF) gives the probability of exactly one value:

P(X = k)

. The cumulative distribution function (CDF) sums these probabilities up to and including k:

F(k) = \sum_{i \leq k} P(X = i)

. The PMF appears as individual spikes or bars showing probability at each point. The CDF shows accumulated probability as a step function. You can recover the PMF from the CDF by taking differences:

P(X = k) = F(k) - F(k-1)

, which equals the height of each step.

Use PMF when asking about exact values. Use CDF when asking about ranges or "at most" probabilities.

For detailed comparison of probability functions including when to use each, see probability mass function vs cumulative distribution function.

Related Probability Tools and Concepts

Discrete Distribution Calculators - Compute exact probabilities, means, and variances for all six distributions with parameter input.

PMF Visualizers - Display probability mass functions as bar charts showing individual probabilities rather than cumulative values.

Continuous Distribution CDFs - Explore cumulative distribution functions for continuous random variables where CDFs are smooth curves rather than step functions.

Probability Mass Function Theory - Understand the mathematical foundation of discrete probability functions and their properties.

Random Variables - Learn the fundamental concept underlying all probability distributions and how they map outcomes to numerical values.

Discrete Distributions Overview - Comprehensive guide to all discrete probability distributions including when to use each type.