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).
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.