Question

Consider the probability distribution of a random variable x. Is the expected value of the distribution necessarily one of the possible values of x? Explain and give examples.

WRITE IN TEXT NOT IN IMAGE TEXT SINCE I CANT READ SOME OF THE TEXT IN IMAGE

Answer 1

In general the expected value of a random variable X is not equal to one of the values of X.

For example consider a random variable X having Uniform Distribution over the set {1,2,3,4,5,6} i.e. the probability distribution of X is

X=x	1	2	3	4	5	6
P[ X=x]	1/6	1/6	1/6	1/6	1/6	1/6

So, X takes the values 1,2,3,4,5,6 with equal probability 1/6 and its expected value is

E(X) = 1*1/6+2*1/6+3*1/6+4*1/6+5*1/6+6*1/6 = 3.5 which is not equal to one of the X values.

In general the Expected value of X should always lie within the maximum and minimum values of X.