Sometimes, probability can mess up your head. You will see a number of values with different probabilities, therefore, interpreting with them becomes difficult. That is why the concept of distribution function was born. This mathematical expression shows the chances of occurring of a specific value or a set of values. Suppose that the X is a discrete random variable with values in \mathbb{R}. These values are sorted from lowest to highest, the distribution function of the variable X is:

 

f\left( x \right) =P(X\le x)

 

The distribution function associates the cumulative probability of each value of the random variable. In addition, the distribution function is also commonly referred to as the cumulative distribution function. You must be wondering why the distribution function is important? Many statisticians use the distribution function because it makes sense for any sort of random variable. It doesn't matter whether the distribution is mixed, continuous, or discrete, it will help them to understand that function properly. Furthermore, the distribution function determines the distribution of the X.

Properties

Below are the properties of cumulative distribution function:

  1. 0\le f\left( x \right) \le 1,\quad -\infty <x<\infty
  2. f\left( -\infty \right) =0,\quad f\left( \infty \right) =1
  3. In case, if { x }_{ 1 }<{ x }_{ 2 } that means f\left( { x }_{ 1 } \right) \le f\left( { x }_{ 2 } \right)
  4. P({ x }_{ 1 }<x\le { x }_{ 2 })=f\left( { x }_{ 2 } \right) -f\left( { x }_{ 1 } \right)

Since we know that we are dealing with probability here, this means that the values of f\left( x \right) will range from 0 to 1. In addition, the value of x can be any real number from -\infty to \infty. In the second property, if we input -\infty as the value of x in the original equation that will be f\left( x \right) =P(X\le -\infty). This is not possible since distribution of the X should always be lesser than x (any real number) and there is nothing smaller than -\infty, therefore, we will say the probability of that occurring is 0. On the other hand, f\left( x \right) =P(X\le \infty), this means distribution of the X is less than infinity, without any doubt, there is nothing greater than infinity and that is why probability of that occurring is 1.

Suppose { x }_{ 1 }=4\quad \& \quad { x }_{ 2 }=9, in this case, the cumulative distribution at { x }_{ 1 } will be less than or it could be equal to cumulative distribution at { x }_{ 2 }. In simple words, as the number increases from left to right, the { x }_{ 2 } function will keep getting bigger or it will stay the same but one thing is for sure, it will never get smaller than { x }_{ 1 }. The last property says, what is the probability of the random variable between two different numbers? To find that you will need to subtract the distributive function of { x }_{ 1 } from the distributive function of { x }_{ 2 } (don't forget that these functions are probabilities and the answer will never exceed from one).

 

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Example

Q.1 Calculate the probability distribution function for the probable scores that can be obtained by throwing a die.

{ x } { p }_{ i }
x<1 0
1\le x<2 \frac { 1 }{ 6 }
2\le x<3 \frac { 2 }{ 6 }
3\le x<4 \frac { 3 }{ 6 }
4\le x<5 \frac { 4 }{ 6 }
5\le x<6 \frac { 5 }{ 6 }
6\le x 1

Representation

 

Q.2  Given the probability distribution for random variable X, find the cumulative distribution and find P\left( x\le 4.5 \right) ,\quad P\left( x>4.5 \right) \quad \& \quad P\left( 1.5<x\le 4.5 \right)

 

x 1 2 4 6
f\left( x \right) 0.2 0.5 0.1 0.2

 

F\left( x \right) =P\left( X\le x \right)

Remember, the small x donates a random number. It means that whatever will be the value of x will be the value of F\left( x \right)

P\left( X\le 4.5 \right) =F(4.5)=0.2+0.5+0.1=0.8

 

P\left( X>4.5 \right) =1-P\left( X\le 4.5 \right) =1-0.8=0.2

 

P\left( 1.5<X\le 4.5 \right) =F(4.5)-F(1.5)

F(1.5)=0.2,\quad \quad F(4.5)=0.8

P\left( 1.5<X\le 4.5 \right) =0.8-0.2=0.6

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.