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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.

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

2 Answers 2

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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 $= 10 + 2.4 = 12.4$

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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?

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