Cumulative Distribution Function

The cumulative Distribution Function (CDF), of a real valued random variable X evaluated at x, is the probability function that X will take a value less than or equal to x. It is used to describe the probability distribution of random variablesA random variable is a variable that defines the possible outcome values of a random phenomenon. It is defined for both discrete and random variables.

In other words, cumulative distribution function finds the cumulative probability for the given value. To determine the probability of a random variable, CDF is used . It is used to compare the probability between values under certain conditions. For discrete distribution functions, CDF gives the probability values till what we specify and for continuous distribution functions, it gives the area under the probability density function up to the given value specified.

Cumulative Distribution Function Formulas

The CDF defined for discrete random variable is given as

\(F_{X}(x)=P(X\leq x)\)

Where X is the probability that takes a value less than or equal to x and that lies in the semi-closed interval (a,b], where a < b.

Therefore the probability within the interval is written as

\(P(a<X\leq b)=F_{X}(b)-F_{X}(a)\)

The CDF defined for continuous random variable is given as

\(F_{X}(x)=\int_{-\infty }^{x}f_{X}(t)dt\)

Here, X is expressed in terms of integral of its probability density function \(f_{X}\)

In case, if the distribution of the random variable X has the discrete component at value b,

\(P(X=b)=F_{X}(b)-\lim_{x\to b}F_{X}(x)\)

Properties of CDF

The cumulative distribution function FX(x) of a random variable has the following important properties:

  • Every CDF \(F_{X}\) is non decreasing and right continuous
\(\large \lim_{x \to-\infty }F_{X}(x)=0 ,\lim_{x \to+\infty }F_{X}(x)=1\)
  • For all real numbers a and b with continuous random variable X, then the function \(f_{X}\) is equal to the derivative of \(F_{X}\), such that
\(F_{X}(b)-F_{X}(a)=P(a<X\leq b)=\int_{a}^{b}f_{X}(x)dx\)

The cumulative distribution function is defined for all real values, sometimes it is defined implicitly rather than defining it explicitly. The CDF is an integral concept of PDF ( Probability Distribution Function )

Consider a simple example for CDF which is given by rolling a fair six sided die , where X is the random variable

We know that the probability of rolling a six sided die is given as:

Probability of getting 1 = P(X≤ 1 ) = 1 / 6

Probability of getting 2 = P(X≤ 2 ) = 2 / 6

Probability of getting 3 = P(X≤ 3 ) = 3 / 6

Probability of getting 4 = P(X≤ 4 ) = 4 / 6

Probability of getting 5 = P(X≤ 5 ) = 5 / 6

Probability of getting 6 = P(X≤ 6 ) = 6 / 6 = 1

From this, it is noted that the probability value always lies between 0 and 1 and it is non-decreasing and right continuous in nature.

Application of CDF

The most important application of cumulative distribution function is used in statistical analysis. In statistical analysis, the concept of CDF is used in two ways.

  • Finding the frequency of occurrence of values for the given phenomena using cumulative frequency analysis.
  • By using empirical distribution function, that uses formal direct estimate of cumulative distribution functions to derive some simple statistics properties.

Sample Example

Question :

The random variable with PDF is given by:

Find the cumulative distribution function(CDF)

Solution :

We know that, \(\int_{-\infty }^{\infty }f(x)dx=1\) \(k\int_{0 }^{1 }(x^{2}+x)dx=1\) \(k\left ( \frac{x^{3}}{3}+\frac{x^{2}}{2} \right )_{0}^{1}=1\) \(k\left ( \frac{5}{6} \right )=1\)

Therefore , k =\(\frac{6}{5}\)

The CDF F(X),is the function of PDF and it can be integrated within the interval \((-\infty ,x)\)

If x is in the interval \((-\infty ,x)\), then

\(F(x)=\int_{-\infty }^{x}f(x)dx\) \(F(x)=\int_{-\infty }^{x}0dx\) \(F(x)=0\)

If x is in the interval [0,1], then

\(F(x)=\int_{-\infty }^{x}f(x)dx\) \(F(x)=\int_{-\infty }^{0}f(x)dx+\int_{0 }^{x}f(x)dx\) \(F(x)=0+\frac{6}{5}\left ( \frac{x^{3}}{3}+\frac{x^{2}}{2} \right )\)

If x is in the interval \((1,\infty )\),then

\(F(x)=\int_{-\infty }^{x}f(x)dx\) \(F(x)=\int_{-\infty }^{0}f(x)dx+\int_{0 }^{1}f(x)dx+\int_{1 }^{x}f(x)dx\) \(F(x)=0+\frac{6}{5}\left (\frac{x^{3}}{3} + \frac{x^{2}}{2} \right )_{0}^{1}+0\) \(F(x)=\frac{6}{5}\times \frac{5}{6}\) \(F(x)=1\)

Therefore the cumulative distribution function CDF is given by,

\(F(x)=\left\{\begin{matrix}0 \ \ ;\ \ \ if\ x<0 \\\frac{6}{5}\left ( \frac{x^{3}}{3}+\frac{x^{2}}{2} \right ) ; if\ 0\leq x\leq 1 \\ 1\ \ \ ;\ if\ x>1 \end{matrix}\right.\)

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Practise This Question

Express 3 × 102 + 2 × 101 + 9 × 100 + 2 × 101 + 5 × 102 as a numeral.