Find the mean and standard deviation for a random variable X

Find the mean and standard deviation for a random variable X having the following CDF:

F(x) = {
0, x < 3
3/8, 3 x < 1
11/16, 1 x < 5
1, x 5

Solution

F(x)=0      x<-3

     =3/8   -3<=x<1

     =11/16 1<=x<5

     =1    x>=5

so here X takes only 3 values... -3,1,5

P[X=-3]=P[X<=-3]-P[X<-3]=F(3)-F(-3-0)=3/8

P[X=1]=P[X<=1]-P[X<1]=F(1)-F(1-0)=11/16-3/8=5/16

P[X=5]=1-P[X=-3]-P[X=1]=1-3/8-5/16=5/16

hence the probability distribution of X is given as---

     X:      -3       1       5

P[X=x]:       3/8     5/16   5/16

therefore mean of X is--

E[X]=-3*3/8+1*5/16+5*5/16=3/4=0.75 [ANSWER]

now E[X²]=((-3)^2)*3/8+(1^2)*5/16+(5^2)*5/16

                =9*3/8+1*5/16+25*5/16=11.5

therefore V(X)= E[X²]-E²[X]11.5-0.75²=10.9375

so standard deviation is suqareroot(10.9375)=3.3072 [answer]