Cumulative Distribution Function#

Definition#

Definition 48 (Cumulative Distribution Function)

Let \(\P\) be a probability function defined over the probability space \(\pspace\).

Let \(X\) be a continuous random variable with sample space \(\S = \R\) and let \(\pdf\) be its probability density function.

Then the cumulative distribution function (CDF) of \(X\) is defined as:

\[ \cdf \lpar x \rpar = \P \lsq X \leq x \rsq = \int_{-\infty}^x \pdf \lpar t \rpar \, dt \]

Properties of CDF#

Proposition 2 (Properties of CDF)

Let \(X\) be a random variable (either discrete or continuous), then the CDF \(\cdf\) of \(X\) satisfies the following properties:

The CDF is non-decreasing.
The CDF is a probability function.

\[ 0 \leq \cdf(x) \leq 1 \]

In particular, we have the minimum of the CDF is 0 and the maximum is 1 for \(x = -\infty\) and \(x = \infty\) respectively.

Proposition 3 (Probability of an Interval)

Let \(X\) be a continuous random variable.

If the CDF \(\cdf\) of \(X\) is continuous at any \(a \leq x \leq b\), then the probability of an interval \([a, b]\) is given by:

\[ \P \lsq a \leq X \leq b \rsq = \cdf(b) - \cdf(a) \]

Definition 49 (Left and Right Continuity)

Let \(X\) be a continuous random variable. Then its CDF \(\cdf\) is said to be [Chan, 2021]:

left continuous if \(\cdf\) is continuous at \(x=b\) if \(\cdf(b) = \cdf(b^{-}) = \lim_{h \to 0} \cdf(b-h)\)
right continuous if \(\cdf\) is continuous at \(x=b\) if \(\cdf(b) = \cdf(b^{+}) = \lim_{h \to 0} \cdf(b+h)\)
continuous if \(\cdf\) is left continuous and right continuous at \(x=b\). This means that

\[ \lim_{h \to 0} \cdf(b-h) = \cdf(b) = \lim_{h \to 0} \cdf(b+h) \]

Theorem 10 (CDF is Right Continuous)

Let \(X\) be a random variable (either discrete or continuous). Then its CDF \(\cdf\) is always right continuous.

\[ \cdf(b) = \cdf(b^{+}) = \lim_{h \to 0} \cdf(b+h) \]

Theorem 11 (Define Probability at a Point)

Let \(X\) be a random variable (either discrete or continuous), then \(\P \lsq X = b \rsq\) is given by

\[\begin{split} \P \lsq X = b \rsq = \begin{cases} \cdf(b) - \cdf(b^{-}) & \text{if } \cdf \text{ is discontinuous at } b \\ 0 & \text{otherwise} \end{cases} \end{split}\]

PDF is Derivative of CDF#

We have seen how we can convert a PDF \(\pdf\) to a CDF \(\cdf\) by integrating the PDF. We now show how to convert a CDF \(\cdf\) to a PDF \(\pdf\) by taking the derivative of the CDF.

Theorem 12 (PDF is Derivative of CDF)

By the Fundamental Theorem of Calculus defined in Theorem 1, given a cumulative distribution function (CDF) \(\cdf\) of a random variable \(X\), we can find its probability density function (PDF) \(\pdf\) by taking the derivative of the CDF:

\[ \pdf \lpar x \rpar = \frac{d}{dx} \cdf \lpar x \rpar = \frac{d}{dx} \int_{-\infty}^x \pdf \lpar t \rpar \, dt \]

if \(\cdf\) is differentiable at \(x\). If \(\cdf\) is not differentiable at \(x=b\), then

\[ \pdf(b) = \P \lsq X = b \rsq = \P \lsq X = b \rsq \delta(x-b) \]

Probability & Statistics

Cumulative Distribution Function

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