The cumulative distribution function has the same definition for a continuous random variable as for a discrete one.

Definition

The cumulative distribution function (CDF) for a continuous random variable \(X\) is the function

\[F(x) \;=\; P(X \le x)\]

This probability can be expressed as an integral,

\[F(x) \;\; = \; \; \int_{-\infty}^x f(t)\;dt\]

Note that this also implies that

\[f(x) \;\; = \; \; \frac {d}{dx} F(x)\]

All cumulative distribution functions monotonically rises from zero to one. However whereas a discrete distribution's CDF is a step function, that of a continuous distribution is a smooth function.

Question: Rectangular distribution

Sketch the cumulative distribution function of a random variable with a rectangular distribution, \(X \sim \RectDistn(1, 5)\).

Question: Exponential distribution

If \(X\) has probability density function

\[ f(x) = \begin{cases} 4\;e^{-4x} & \quad \text{for } x \ge 0\\[0.2em] 0 & \quad \text{otherwise} \end{cases} \]

what is its cumulative distribution function?

(Both solved in full version)