# Expected value for stock movement

The price of a stock has a $$30\%$$ chance to increase by $$\5$$ and a $$70\%$$ chance to decrease by $$\1$$ each day (with independent movement across days). If you purchased a stock for $$\10$$ what is the expected price 3 days later?

Not sure how to approach this problem. I calculated that extremes like this:

Goes up each day for three days: $$.30 \cdot 15 + .30 \cdot 20 + .30 \cdot 25$$

Goes down each day for three days: $$.70 \cdot 9 + .70 \cdot 8 + .70 \cdot 7$$

Then maybe average those?

But not sure if this is an approach.

• Each day is an independent processso try to adapt the linearity of expected value to the random variable X=X1 + X2 + X3 Commented Aug 10, 2022 at 2:29

So the question is simple. Let us say we associate a random variable $$X_i$$ which takes value $$5$$ with probability $$0.30$$ and the value $$-1$$ with probability $$0.7$$.
So, $$X_i$$ denotes the movement of a stock on $$i^{th}$$ day.
Now Expected change in the price of a stock in a single day is given by $$E(X_i) = (-1)(0.7) + (5)(0.3) = -0.7 + 1.5 = 0.8$$
Now since, price change in each day is independent of the price change at other day. So, Expected change of price after $$3$$ days $$= E(X_1) + E(X_2) + E(X_3) = 2.4$$

Hence the expected price of stock after $$3$$ days $$= 10 + 2.4 = 12.4$$

We started with $$C=\10$$.

The price after $$3$$ days will be $$C+\sum_{i=1}^3 X_i$$ where $$X_i$$ is the change of the price.

By linearity of the expectation, the expected price would be $$C+\sum_{i=1}^3 E[X_i]$$.

Hence, to solve the problem, we just have to solve for $$E[X_i]$$. Are you able to take it from here?