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Understanding Markov’s theorem with example

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  • Difficulty Level : Hard
  • Last Updated : 24 Mar, 2021

In this article, we will discuss the overview of the Markov theorem and also will discuss the expression of the Markov theorem, and finally concluded with an example to understand the Markov theorem. Let’s discuss it one by one.

Markov’s Theorem :  
Markov theorem states that if R is a non-negative (means greater than or equal to 0) random variable then, for every positive integer x, Probability for that random variable R to be greater than or equal to that positive integer x is upper bounded by the Expected value of random variable R upon x.

Expression of Markov’s Theorem : 
Mathematically, it can be written as follows.

If R >=0 , then ∀ x >0,
P(R>=x) <= Ex( R ) / x

Points to Remember :
Please note that random variable R has to be non-negative for applying the above Markov’s theorem.

If R is non-negative ∀ C > 0, then
P (R >= c*Ex( R ) ) <= 1/c 

An extended version of Markov’s theorem states the following expression as follows.

If R ≤ U for some U in the set of a real number ( U ∈ IR) then,
∀ x >0,
P(R ≤ x) ≤ (U - Ex( R ) ) / ( U- x )

Example :  
Here, we will discuss the example to understand this Markov’s Theorem as follows. 
Let’s say that in a class test for 100 marks, the average mark scored by students is 75. Then what’s the probability that a random student picked from the class has less than 0r equal to 50 marks.

To solve this, let’s define a random variable R = Score of a random student. Since R is upper bounded by 100, so we use the extended version of the Markov theorem as discussed above.

Now, by using the given below expression of Markov’s Theorem, we will solve this problem as follows.

Expression :
If R >=0 , then ∀ x >0,
P(R>=x) <= Ex( R ) / x 
So, U = 100,
Ex ( R ) = 75
then, use the above formulae,
P (R <= 50 ) = (  100-  75) / ( 100- 50  ) = 25/ 50 = 1/2
which gives the answer as 0.5 

So, the probability that a random student’s score is almost 50 is upper bounded by 0.5 

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