How to calculate the expected value of a discrete random variable?

Question

A private investor has capital of £16,000. He divides this into eight units of £2,000, each of which he invests in a separate one-year investment. Each of these investments has three possible outcomes at the end of the year:

1. total loss of capital probability 0.1
2. capital payment of £2,000 probability 0.7
3. capital payment of £5,000 probability 0.2

The investments behave independently of one another, and there is no other

return from them.

Calculate the expected payment received by the investor at the end of the year.

I am unable to figure out how to create a probability distribution from the given data.

If I could create it,then I think I can calculate it just applying the formula.

Answer 1

Hint: Independence is not necessary. For $i=1$ to $8$, let $X_i$ be the payment from investment $i$. Then $Y=X_1+\cdots +X_8$ is the total payment.

By the linearity of expectation, we have $E(Y)=E(X_1)+\cdots+E(X_8)$.

I think you know how to compute $E(X_i)$. Note that we do not need to find the distribution of random variable $Y$.

Answer 2

You can rather see each split of 2000 pounds as having it's own distribution. From stats we know the result the expectation of the sum is the sum of the expectation as it is a linear operator. So you would then calculate the sum of each expectation from each 2000 pounds.