Question

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 last 3 years. The pmf of Y is the following: 1 0.20 2 0.30 0.05 y 0 P(V) 0.45 (a) Compute E(Y). E(Y) = (b) Suppose an individual with Y violations incurs a surcharge of $80V. Calculate the expected amount of the surcharge,

Answer 1

Answer:

a) $E(Y)= \sum_{i=1}^n Y_i P(Y_i)$

And replacing we got:    

$E(Y) = 0*0.45 +1*0.2 +2*0.3 +3*0.05= 0.95$

b) $E(80Y^2) =80[ 0^2*0.45 +1^2*0.2 +2^2*0.3 +3^2*0.05]= 148$

Step-by-step explanation:

Previous concepts

In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".  

The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).  

And the standard deviation of a random variable X is just the square root of the variance.  

Solution to the problem

Part a

We have the following distribution function:

Y        0         1         2       3

P(Y)  0.45    0.2    0.3   0.05

And we can calculate the expected value with the following formula:

$E(Y)= \sum_{i=1}^n Y_i P(Y_i)$

And replacing we got:    

$E(Y) = 0*0.45 +1*0.2 +2*0.3 +3*0.05= 0.95$

Part b

For this case the new expected value would be given by:

$E(80Y^2)= \sum_{i=1}^n 80Y^2_i P(Y_i)$

And replacing we got

$E(80Y^2) =80[ 0^2*0.45 +1^2*0.2 +2^2*0.3 +3^2*0.05]= 148$