What is a Cumulative Distribution Function (CDF)?

The cumulative distribution function, or cdf, of a random variable tells us the probability that it is less than or equal to any chosen value.

For a random variable $X$, we write

$$F_X(t)=\mathbb{P}(X\leq t).$$

The word cumulative is helpful here: the cdf collects up (or “accumulates”) all the probability up to and including $t$.

For example, suppose $Y$ is the result of rolling a fair die. Then

$$F_Y(3)=\mathbb{P}(Y\leq 3)=\frac{3}{6}=\frac{1}{2}.$$

We get this by adding the probabilities of rolling 1, 2 or 3.

More generally, for a discrete random variable,

$$F_X(t)=\mathbb{P}(X\leq t)=\sum_{x\leq t}\mathbb{P}(X=x).$$

Importantly, the cdf is defined for every real value of $t$, not just values that $X$ itself can take. For the fair die, for instance,

$$F_Y(3.7)=\frac{1}{2},$$

since the possible values less than or equal to 3.7 are still just 1, 2 and 3. The cdf of a discrete random variable typically looks like a step function.

Now suppose that $X$ is continuous, with probability density function $f_X(x)$. Recall that probabilities are then given by areas under the pdf.

So $F_X(t)=\mathbb{P}(X\leq t)$ is simply the total area under the pdf to the left of $t$:

$$F_X(t)=\int_{-\infty}^{t}f_X(x),dx.$$

This is the same idea as in the discrete case. We are still accumulating probability up to a chosen value; we are just using an integral rather than a sum.

For continuous random variables, the cdf and pdf are therefore closely related. At points where the cdf is differentiable, we can recover the pdf by differentiating:

$$f_X(x)=\frac{dF_X(x)}{dx}.$$

So, whether $X$ is discrete or continuous, the definition of the cdf is always the same:

$$F_X(t)=\mathbb{P}(X\leq t).$$

Only the way we calculate this probability changes.

Since a cdf gives probabilities, it must always satisfy

$$0\leq F_X(t)\leq 1.$$

As $t$ moves further to the right, more probability is accumulated, so the cdf can only stay the same or increase.

Background:

• Probability Density Functions