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.


2 Answers 2


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 $Y$.

  • $\begingroup$ Search for 'inverse CDF method' on this site and elsewhere. $\endgroup$
    – BruceET
    Nov 2, 2015 at 6:30
  • $\begingroup$ how to calculate the E(X1) from the data given? $\endgroup$
    – RajSharma
    Nov 2, 2015 at 6:46
  • $\begingroup$ $E(X_1)=(0.1)(0)+(0.7)(2000)+(0.2)(5000)$. $\endgroup$ Nov 2, 2015 at 6:56
  • $\begingroup$ thanks... I solve this problem using this approach. $\endgroup$
    – RajSharma
    Nov 2, 2015 at 12:43

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.


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