Lesson 25 – More expectation

My old high school friend is now a successful young businessman. Last week, he shared thoughts on one of his unusual stock investment schemes. Every few months, he randomly selects four stocks from four different sectors and bets equally on them. I asked him for a rationale. He said he expects two profit making stocks on average assuming that the probability of profit or loss for a stock is 0.5. I think since he picks them at random, he also assigns a 50-50 chance of win lose.

My first thought

This made a nice expected value problem of the sum of random variables.

The expected number of profit making stocks in his case is 2. We can assign X1, X2, X3, and X4 as the random variables for individual stocks with outcomes 1 if it makes a profit and 0 otherwise. We can assign Y as the total number of profit making stocks; ranging from 0 to 4. His possible outcomes are:

As we can see, the total number of profit making stocks in these scenarios are 4, 3, 3, 2, 3, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0. The average of these numbers is 2; the expected number of profit making stocks.

Another way of getting at the same number is to use the expected value formula we learned in lesson 24.

`E[Y] = 4(1/16) + 3(4/16) + 2(6/16) + 1(4/16) + 0(1/16) = 2`

An important property of expected value of a random variable is that the mean of the linear function is the linear function of the mean.

Y = X1 + X2 + X3 + X4

Y is another random variable that comes from a combination of individual random variables. For sum of random variables,

`E[Y] = E[X1] + E[X2] + E[X3] + E[X4]`

Detailed folks can go over the derivation.

Now, E[X1] = 0.5(1) + 0.5(0) = 0.5, since the outcomes are 1 and 0 and the probabilities are 0.5 each. Adding all of them, we get 2.

So you see, the additive property makes it easy to estimate the expected value of the sum of the random variables instead of writing down the outcomes and computing the probability distribution of Y.

Other simple rules when there are constants involved;

Try to derive them as we did above.

My second thought

Stock market might be more complicated than a coin flip experiment. But what do I know; he clearly has more net worth than me. I guess since this is only one of his investment schemes, he is just playing around with his leftovers.

My final thought

I am only good for teaching probability; not using it like him. But again, most ivory tower professors only preach; don’t practice. Hey, they are protected. Why should they do real things?

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