Cumulative Distribution Function

In Mathematics, Statistics and Probability play a very important role in helping to calculate data sufficiency. The Cumulative Distribution Function is a major part of both these sub-disciplines and it is used in a number of applications. This function, also abbreviated as CDF, takes into account that a random variable valued at a real point, like X, is evaluated at x. In this case, the function holds that X will be of a lower value than x or will be valued the same as x. 

We mentioned that X is a random variable. What this means is that this variable explains the probable resulting values on an unexpected phenomenon. Understanding this is fundamental to understanding the Cumulative Distribution Function. 

What is a Cumulative Distribution Function?

CDF of a random variable ‘X’ is a function which can be defined as,

FX(x) = P(X ≤ x)

The right-hand side of the cumulative distribution function formula represents the probability of a random variable ‘X’ which takes the value that is less than or equal to that of the x. The semi-closed interval in which the probability of ‘X’ lies is (a.b], where a < b, 

P(a < X ≤ b) = F_x(b) - F_x(a)

Note that the ≤ sign which is used here is not conventionally used at all times, but it can be useful for discrete distributions. Depending on this, the right use of binomial and Poisson’s Distribution tables are employed. Many important formulas in Mathematics are totally dependent on the equal to or the lesser than sign, such as Paul Levy’s inversion formula. 

Understanding Cumulative Distributions

When random variables such as X, Y, and so on are solved, the letter that is used to subscript is the lower case of the same letter. This is done to avoid unnecessary confusion and mixups. However, the use of a subscript may not be necessary when a single random variable is being used. If the capital letter F is used for the cumulative distribution function then the lowercase letter f is used in the probability density and the probability mass functions. 

The continuous random variable probability density function can be derived by differentiating the cumulative distribution function. This is shown by the Fundamental Theorem of Calculus. 

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

The CDF of a continuous random variable ‘X’ can be written as integral of a probability density function. The ‘r’ cumulative distribution function represents the random variable that contains specified distribution.

\[F_x(x) = \int_{-\infty}^{x} f_x(t)dt \] 

Understanding the Properties of CDF

In case any of the below-mentioned conditions are fulfilled, the given function can be qualified as a cumulative distribution function of the random variable: 

• Every CDF function is right continuous and it is non increasing. Where \[\lim\limits_{x \rightarrow -\infty } F_x(x) = 0,  \lim\limits_{x \rightarrow +\infty } F_x(x) = 1 \] 

• If ‘X’ is a discrete random variable then its values will be x1, x2, .....etc and the probability Pi = p(xi) thus the CDF of the random variable ‘X’ is discontinuous at the points of xi. FX(x) = P(X ≤ x) = Σxi ≤ x P(X = xi) = Σxi ≤ x p(xi). 

• If the CDF of a real-valued function is said to be continuous, then ‘X’ is called a continuous random variable F_x(b) - F_x(a) = P(a < X ≤ b) = ∫ab fX(x) dx.

The function fx = derivative of Fx is the probability density function of X.

Derived Functions

• Complementary Cumulative Distribution Function: It is also known as tail distribution or exceedance, it is defined as, F_x(x)=P(X>x)=1−FX(x)

• Folded Cumulative Distribution: When the cumulative distributive function is plotted, and the plot resembles an ‘S’ shape it is known as FCD or mountain plot.

• Inverse Distribution Function: The inverse distribution function or the quantile function can be defined when the CDF is increasing and continuous. F−1(p),pϵ[0,1]F−1(p),pϵ[0,1]F^{-1} (p), p \epsilon [0,1] such that F(x) = p.

• Empirical Distribution Function: The estimation of cumulative distributive function that has points generated on a sample is called empirical distribution function.

Solved Example 1. 

1. What is the cumulative distribution function formula? 

Given the CDF F(x) for the discrete random variable X, 

Find: (a) P(X = 3) (b) P(X > 2)

Solution: 

CDF of a random variable ‘X’ is a function which can be defined as,

FX(x) = P(X ≤ x)(a) P(X = 3)

To obtain the CDF of the given distribution, here we have to solve till the value is less than or equal to three. From the table, we can obtain the value 

F(3) = P(X  3) = P(X = 1) + P(X = 2) + P(X = 3)

From the table, we can get the value of F(3) directly, which is equal to  0.67.

(b) P(X > 2)

P(X > 2) = 1 - P(X ≤ 2)

P(X > 2) = 1 - F(2)

P(X > 2) = 1 - 0.32P

(X > 2) = 0.87

2. What is the CDF of normal distribution in r?

Given the probability distribution for a random variable x, 

find (a) P(x ≤ 4.5) (b) P(x > 4.5)

Solution: 

The CDF of the normal distribution can be denoted by " φ " the probability of a random variable that has a related error function.

(a) P(x ≤ 4.5) = F(4.5) = 0.8

(b) P(x > 4.5) = 1 - P(x ≤ 4.5)

(c) P(x > 4.5) = 1 - 0.8

(d) P(x > 4.5) = 0.2