cumulative distribution function

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable $X$ is a function that gives the probability that $X$ will take a value less than or equal to $x$. It is defined as:

$$\begin{array}{rl}{F}_X(x)& =P(X\le x)={\int }_{-\infty }^x{f}_X(t)dt\end{array}$$

where $f_X(t)$ is the probability density function (PDF) of $X$.

standard normal distribution CDF ($\Phi$)

For the standard normal distribution ($\mu =0,\sigma =1$), the CDF is often denoted as $\Phi (x)$.

For the interval $[-1,1]$, which represents 1 standard deviation from the mean:

$$P(-1\le X\le 1)=\Phi (1)-\Phi (-1)\approx 0.8413-0.1587=0.6826$$

This shows that approximately 68% of values drawn from a standard normal distribution lie within one standard deviation of the mean.

The CDF is a non-decreasing function with a range of $[0,1]$

For any real-valued random variable, the CDF has the following properties:

• ${\lim }_{x\to -\infty }{F}_X(x)=0$
• ${\lim }_{x\to \infty }{F}_X(x)=1$
• $F_X(x)$ is non-decreasing (i.e., $F_X(a)\le F_X(b)$ for all $a\le b$)

Relationship between PDF and CDF

The PDF is the derivative of the CDF (when the derivative exists):

$$f_X(x)=\frac{d}{dx}{F}_X(x)$$

Conversely, the CDF is the integral of the PDF:

$$F_X(x)={\int }_{-\infty }^x{f}_X(t)dt$$