THX A random variable Y can only take values in 1 0 1 The ex

THX

A random variable Y can only take values in {-1, 0, 1}. The expected value of Y is 0 and its variance is 1/2. Find the probability distribution of Y.

Solution

Let, P(-1)=a,P(0)=b,P(1)=c

HEnce, E(Y)=0

a*-1+b*0+c*1=0

Hence, a=c

Var(Y)=E(Y^2)=1/2=a+c=1/2

Hence, a=c=1/4

Also

a+b+c=1

Hence,b=1/2

So this gives the probability distribution

P(-1)=P(1)=1/4

P(0)=1/2