An individual who has automobile insurance from a certain company is randomly selected. Let Y be the number of moving violations for which the individual was cited during the past 3 years. The pmf of Y is: y 0 1 2 3
P(y) 0.50 0.25 0.20 0.05

Required:
a. Compute E(Y) and V(3Y−10)
b. Suppose an individual with Y violations incurs a surcharge of $(95Y2−25). Calculate the expected amount of the surcharge.
c. What is the standard deviation of surcharges?

a.

E(y)=0.8

V(3y-10)=7.74

b.

The expected amount of surcharge is 117.5.

c.

The standard deviation of surcharges is 217.6723.

Step-by-step explanation:

a.

y          0      1      2    3

p(y)    0.5 0.25 0.2 0.05

y*p(y)  0    0.25 0.4 0.15

E(y)=sum(y*p(y))=0+0.25+0.4+0.15

E(y)=0.8.

y²          0      1      4    9

y²p(y)   0 0.25 0.8 0.45

V(y)=E(y²)-[E(y)]²

E(y²)=sum(y²*p(y))=0+0.25+0.8+0.45=1.5

V(y)=E(y²)-[E(y)]²

V(y)=1.5-[0.8]²

V(y)=1.5-0.64

V(y)=0.86

V(3y-10)=9V(y)=9*0.86=7.74

V(3y-10)=7.74

b.

E(95Y²-25)=95E(Y²)-25=95*1.5-25=142.5-25=117.5

E(95Y²-25)=117.5

The expected amount of surcharge is 117.5.

c.

SD(95Y²-25)=?

SD(95Y²-25)=√V(95Y²-25)

V(95Y²-25)=95²*V(Y²)

V(95Y²-25)=9025*V(Y²)

V(Y²)=E[(Y²)²]-[E(Y²)]²

y^4          0      1      16    81

y^4*p(y)   0  0.25   3.2 4.05

E[(Y²)²]=sum(y^4*p(y))=7.5

V(Y²)=E[(Y²)²]-[E(Y²)]²

V(Y²)=7.5-[1.5]²

V(Y²)=7.5-2.25

V(Y²)=5.25

V(95Y²-25)=9025*V(Y²)

V(95Y²-25)=9025*5.25

V(95Y²-25)=47381.25

SD(95Y²-25)=√47381.25

SD(95Y²-25)=217.6723

The standard deviation of surcharges is 217.6723.

walter needs to make at least 5 more cookies but no more than 40. 5 ≥ x ≥ 40;

step-by-step explanation:

walter has to make a minimum of 20 cookies. he already has 15 cookies made, so he must make 5 more to meet the minimum.

the maximum amount of cookies he can make is 55. he already has 15, so he has to make no more than 40 more cookies to meet the maximum.

4 and 5)16.97 or 12√2

6)15.59 or 9√3

7)y = 15.59 or 9√3     x =27

8)17.80