A cumulative distribution function, or CDF, shows how probability builds up as you move from left to right along a number line. For a random variable X, the CDF gives the probability that X is less than or equal to a chosen value x. It is one of the most useful ways to describe a probability distribution because it works for both discrete and continuous variables.
CDFs matter because they let you read probabilities over intervals directly from a graph or formula.
A CDF is built by adding up probabilities for a discrete distribution or by accumulating area under a density curve for a continuous distribution. For a probability mass function, the CDF jumps at values that can actually occur, creating a step-shaped graph. For a probability density function, the CDF increases smoothly because probability is spread continuously across intervals.
Once you know the CDF F(x), probabilities such as P(a < X <= b) can be found by subtracting F(a) from F(b).
Understanding Statistics: Cumulative Distribution Functions
A graph of accumulated probability has a useful shape. It begins near zero far to the left and ends near one far to the right. It can stay flat over ranges where no outcomes occur.
It can never move downward, because including more possible outcomes cannot remove probability already counted. In a discrete model, the size of a jump tells you the chance of the value at that jump. This gives an important detail about endpoints.
At a jump, the graph includes the probability of that exact value. Just before the jump, it does not. For a continuous model, a single exact value has probability zero, so this endpoint detail does not change an interval probability.
The steepness of a smooth cumulative graph carries information about where values are concentrated. A steep section means probability is gathering quickly over a short range. A nearly flat section means that range is unlikely.
Students often confuse the height of a density curve with probability. Density height is not itself a probability. Probability comes from the area over an interval.
The cumulative graph avoids some of this confusion because its vertical scale is already probability, from zero to one. Its slope is linked to density, but only where the graph is smooth. A sharp corner or flat section needs careful interpretation rather than a simple slope calculation.
Cumulative graphs are especially useful for finding percentiles. The ninety percent point is a value where the accumulated probability reaches ninety percent. This can describe a test score, a rainfall total, or the time until a bus arrives.
A median is a middle value based on the fifty percent level. In a continuous distribution, one can often find a percentile by moving horizontally from a chosen probability, then down to the number line. In a discrete distribution, the target level may fall inside a jump.
The usual choice is the smallest value whose accumulated probability reaches or passes the target. This explains why some percentiles are whole numbers or may have more than one reasonable description.
Data from a real class or experiment can produce an empirical cumulative distribution. First, sort the observations from smallest to largest. Then increase the running proportion by one divided by the sample size whenever an observation is reached.
The result is a step graph that shows the sample directly, without assuming a bell shape or another model. It helps compare two groups, such as travel times from different routes or heights from different age groups.
When reading any cumulative graph, check the scale on both axes, notice jumps, and identify whether a requested interval includes its endpoints. Small mistakes with left and right endpoints can matter greatly when outcomes are discrete.
Key Facts
- The cumulative distribution function is F(x) = P(X <= x).
- For a discrete random variable, F(x) = sum of P(X = k) for all k <= x.
- For a continuous random variable, F(x) = integral from -infinity to x of f(t) dt.
- Interval probability is P(a < X <= b) = F(b) - F(a).
- Every CDF is nondecreasing, with values between 0 and 1.
- For a continuous distribution with PDF f(x), f(x) = dF/dx where the derivative exists.
Vocabulary
- Cumulative distribution function
- A function F(x) that gives the probability that a random variable is less than or equal to x.
- Probability mass function
- A function that gives the probability of each possible value of a discrete random variable.
- Probability density function
- A function whose area over an interval gives the probability that a continuous random variable falls in that interval.
- Discrete random variable
- A random variable that can take separate, countable values such as 0, 1, 2, or 3.
- Continuous random variable
- A random variable that can take any value in an interval, such as height, time, or temperature.
Common Mistakes to Avoid
- Treating a PDF value as a probability is wrong because probability for a continuous variable comes from area under the curve, not the height of the curve at one point.
- Forgetting the less than or equal to in F(x) = P(X <= x) is wrong because discrete CDFs include the probability at x itself, which creates jumps.
- Subtracting in the wrong order for interval probabilities is wrong because P(a < X <= b) must be F(b) - F(a), not F(a) - F(b).
- Drawing a discrete CDF as a smooth curve is wrong because probability is added only at allowed values, so the graph should stay flat between jumps.
Practice Questions
- 1 A discrete random variable X has P(X = 0) = 0.20, P(X = 1) = 0.50, and P(X = 2) = 0.30. Find F(0), F(1), and F(2).
- 2 A random variable has CDF values F(2) = 0.35 and F(5) = 0.82. Find P(2 < X <= 5).
- 3 Explain why the CDF of a fair six-sided die is step-shaped, while the CDF of a normally distributed measurement is smooth.