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Cumulative Distribution Function

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Introduction to Cumulative Distribution Function

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Probability deals with the measure of the occurrence of an event and statistics deal with the collection, organization, and analysis of the data. Cumulative distribution function or CDF distribution is of a random variable ‘X’ is evaluated at ‘x’, where the variable ‘X’ takes the value which is less than or equal to the ‘x’. In the scalar continuous distribution, the area that is present under the probability density function can be found, which is found from the negative infinity to ‘x’. CDF is also used to specify the probability distribution of multivariate random variables.

Let us discuss what is cumulative distribution function, its formula, properties, its applications, and examples.


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) = FX(b) - FX(a)

Here “≤” in the definition is a sign convention that is not used conventionally but it is found to be important for the discrete distributions. Depending on this sign convention the proper usage of binomial and Poisson's distribution tables are used properly. Some important formulas such as Paul Levy’s inversion formula are completely based on the “less than or equal to” sign.


Cumulative Distributions

While solving for several random variables such as X, Y… etc, the corresponding letter is used in the subscript of the functions to avoid confusion, whereas solving for a single random variable the usage of subscript can be avoided. Capital ‘F’ is used for cumulative distributive function and lower case ‘f’ is used for probability density function and probability mass function.

By differentiating the cumulative distribution function, the continuous random variable probability density function can be obtained, which was done by the usage of the Fundamental Theorem of Calculus.

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

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.

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


CDF Distribution - Properties:

If any of the function satisfies the below-mentioned properties of a CDF distribution then that function is considered as the CDF of the random variable:

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

  • 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. \[F_{X}(x) = P(X \leq x) = \sum_{x_{i} \leq x} P(X = x_{i}) = \sum_{x_{i} \leq x} p(x_{i})\]

  • If the CDF of a real-valued function is said to be continuous, then ‘X’ is called a continuous random variable \[F_{x}(b) - F_{x}(a) = P(a < X \leq b) = \int_{a}^{b} f_{x}(x) dx\]

The function fX = derivative of FX is the probability density function of X.


Derived Functions:

  • Complementary Cumulative Distributive Function: It is also known as tail distribution or exceedance, it is defined as,

\[\overline{F_{X}(x)} = P(X > x) = 1 - F_{X}(x)\]

  • 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. 

\[F^{-1} (p), p \epsilon [0,1]\] 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.


Cumulative Standard Normal Distribution

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


(Images will be uploaded soon)


Solved Example

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)


x

1

2

3

4

5

F(x)

0.2

0.32

0.67

0.9


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 - 0.32

P(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)

P(x > 4.5) = 1 - 0.8

P(x > 4.5) = 0.2


Conclusion

The concept of CDF is used in statistical analysis in two ways. The frequency of the values that occur less the reference value is referred to as cumulative frequency analysis. Two statistical hypothesis tests provide evidence about the sample data that was derived from a given distribution. The Kolmogorov–Smirnov test is to check if the empirical data is different from the ideal distribution. The Kuiper's test is used when the domain of distribution is cyclic.

FAQ (Frequently Asked Questions)

1. What is the Cumulative Distribution Function?

Ans: CDF of a random variable ‘X’ is defined as a function given by, FX(x) = P(X ≤ x)where the x ∈ R. This indicates that CDF is applicable for all the x ∈ R. It helps to calculate the probability of a random variable where the population is taken less than or equal to a particular value.

2. Write Down the Properties of CDF.

Ans: The properties of CDF are as follows,

  • Every CDF function is right continuous and it is non increasing. Where limx ⟶ -∞ FX(x) = 0, limx ⟶ +∞ FX(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 FX(b) - FX(a) = P(a < X b) = ∫ab fX(x) dx.