Estimation of probability

Question

Let the random variable X be the number of months that a certain owner of an estate needs to pay a loan in the Bank if he/she has a contract with insurance company to help him/her to pay a loan.

Say X has the p.m.f

f(x)=5-x /10' x= 1,2,3,4

If the owner of an estate is to receive from an insurance company $200 for each of the first two months in the Bank and $100 for each month after the first two months.what is the expected payment for the loan in the Bank?

Please give all the details!

Answer 1

Let Y be the random variable denoting the amount of the payment in dollars for the loan. Then

Y = (1)(200) = 200 if X = 1,

Y = (2)(200) = 400 if X = 2,

Y = (2)(200) + (1)(100) = 500 if X = 3,

Y = (2)(200) + (2)(100) = 600 if X = 4.

Thus, the PMF of Y is

P(Y=y) = (5-1)/10 = 4/10 if Y = 200

P(Y=y) = (5-2)/10 = 3/10 if Y = 400

P(Y=y) = (5-3)/10 = 2/10 if Y = 500

P(Y=y) = (5-4)/10 = 1/10 if Y = 600

Therefore, the expected value of Y is

E[Y] = (4/10)(200)+(3/10)(400)+(2/10)(500)+(1/10)(600) = 360.

The expected payment is $360.