Question

3. Let X be a discrete random variable with the following PMF: 0.1 for x 0.2 for 0.2 for x=3 Pg(x)=〈 0.1 for x=4 0.25 for x=5 0.15 for x=6 otherwise a) (10 points) Find E[X] b) (10 points) Find Var(X) c) Let Y-* I. (15 points) Find E[Y] II. (15 points) Find Var(Y) X-HX 4. Consider a discrete random variable X with E [X]-4x and Var(X) = σ. Let Y a. (10 points) Find E[Y] b. (20 points) Find Var(Y)

Answer 1

3) from given pmf:

x	P(x)	xP(x)	x2P(x)
1	0.1	0.1	0.1
2	0.2	0.4	0.8
3	0.2	0.6	1.8
4	0.1	0.4	1.6
5	0.25	1.25	6.25
6	0.15	0.9	5.4
total	1	3.65	15.95

a)E(X) =$\sum$xP(x) =3.65

b) E(X²) =$\sum$x²P(x) =15.95

hence Var(X) = E(X²)-(E(X))² =15.95-3.65² =2.6275

c) from Y=2/X ;

below is distribution of Y:

y	P(y)	yP(y)	y2P(y)
2.000	0.100	0.200	0.400
1.000	0.200	0.200	0.200
0.667	0.200	0.133	0.089
0.500	0.100	0.050	0.025
0.400	0.250	0.100	0.040
0.333	0.150	0.050	0.017
total	1	0.7333	0.7706

i) E(Y) =0.7333

ii) Var(Y) =0.7706-0.7333² =0.2328

4)

E(Y) =E((X- $\mu$_x)/$\sigma$_x) =(1/$\sigma$_x)*(E(X)-E( $\mu$_x)) =(1/$\sigma$_x)*($\mu$_x-$\mu$_x) =0

Var(Y) =Var((X- $\mu$_x)/$\sigma$_x) =(1/$\sigma$_x)*(Var(X)+Var( $\mu$_x)) =(1/$\sigma$_x)*$\sigma$_x =1