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If I have 100,000 dollars to invest in stocks, and I can invest 1000 dollars in any particular stock my profit will be 200, 100, 0 and -100 dollars with probability .25 each. There are 100 different stocks to choose from and they all behave independently of each other.

a) How do I find the expected value if I invest 100,000 dollars in one stock and

b) If I invest 1000 in 100 different stocks?

What I did was $$E(X)=100[200*.25+100*.25+0*.25-100*.25]=5000$$

However the actual problem is asking about the probability of profit being 8000 dollars or more and in order to do that I'm assuming the distribution is normal so I'm looking for the $Var(X)$ so I can get the standard deviation. My $E(X^2)-(E(X))^2)$ isn't adding up because im getting a higher $(E(X))^2$ than I am $E(X^2)$. So my expected value must be wrong. Or am I doing something else wrong?

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Your expected value is fine. If I had to guess, you are squaring the $100$ when you do $E(X)^2$ and not when you do $E(X^2)$, but you don't show your calculation for $E(X^2)$. For a single stock, the $E(X)=50, E(X^2)=15000, E(X^2)-E(X)^2=12500$, nicely positive.

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