Question

Suppose that ? is a random variable with the following pmf: ?(?) = ?/ ? ; ? = 1,2,3,4,5

a) Find the expected value and variance of ?.

b) Find the expected value of √?, ?[√?].

Answer 1

For a valid pdf sum of probabilities must be equal to 1. So

a)

The expected value is

------------------

Now,

So variance of Y will be

b)

Following table shows the calculations:

Y	sqrt(Y)	P(sqrt(y))=P(Y=y)	P(sqrt(y))	sqrt(y)*P(sqrt(y))
1	1	1/15	0.066666667	0.066666667
2	1.414213562	2/15	0.133333333	0.188561808
3	1.732050808	3/15	0.2	0.346410162
4	2	4/15	0.266666667	0.533333333
5	2.236067977	5/15	0.333333333	0.745355992
Total				1.880327962

So,