Markov's theorem with example

Markov's theorem is a concept from probability theory that helps us understand and make predictions about random processes. It's especially useful in situations where we want to estimate the likelihood of certain outcomes based on what we already know. Markov's Theorem is commonly used to calculate an upper bound on the probability that a random variable takes a value greater than a certain threshold. This is particularly useful in fields like finance, economics, and computer science

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, the 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.

Markov's theorem gives us a way to predict the "worst-case" scenario, which helps in areas like risk management and system optimization.

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 the random variable R has to be non-negative to apply 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:  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 or equal to 50 marks.

Solution:

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