 Dear students, let me present to you an interesting theorem followed by its proof. The theorem is as follows, if the variance of a random variable X exists, then the expected value of X square is greater than or equal to the square of the expected value of X. As you can see on the screen, E of X square is greater than or equal to E of X whole square. So, how do we prove this one? Let us start from the variance. What is the basic definition of the variance? The variance of X is given by the expected value of X minus mu whole square. Because mu is the same thing as E of X. So, a square minus 2ab plus b square, we open it in the same way. So, therefore, this expression will be equal to the expected value of X square minus 2x into E of X plus E of X whole square. So, what will we have? It will be equal to the expected value of the first term minus the expected value of the second term plus the expected value of the third term. So, what is it? The expected value of the X square, that is the first term, minus 2 times expected value of X into expected value of X plus the expected value of X whole square. Therefore, expected value of a constant is equal to that constant. All right, what do we get? Expected value of X square minus 2 times expected value of X whole square plus one time expected value of X whole square. And what do you get finally? E of X square minus E of X whole square. That is equal to what we have just now achieved, E of X square minus E of X whole square. So, now this equation, you note that the E of X minus E of X whole square, so square of any quantity that can never be negative. The square of any quantity either would be positive and if that quantity itself is zero, then the square will also be zero. So, because the thing that is equal to our E of X square minus E of X whole square, that thing itself is expected value of an X square, so that cannot be negative. So, therefore, the final conclusion is that we have arrived at the point where we are saying that E of X square minus E of X whole square is greater than or equal to zero. And you get what you wanted to prove that E of X square is greater than or equal to E of X whole square. So, this is a basic property and it's quite simple.