So I have this question and can't get the final result –
If a stock goes up daily by $X$ dollars, and the distribution is $X∼U(−0.15,0.37)$. The cost of a stock today is one dollar and I want to buy $200$ stocks.
What is the probability that in $75$ days the value of all stocks in my hand ($200$ stocks) will not go above $1800$ dollars?
I multiplied 200 by the uniform distribution commutative formula, with $X=1$.
and then I was not sure how to calculate the value of stock after $75$ days so I calculated it with CDF also, as $g(x)$, am I not in the right direction?
Best Answer
First of all you have to evaluate the expected value. $\mathbb E(X)=\frac{0.37+(-0.15)}{2}=0.11$. And the variance is $Var(X)=\frac{(0.37-(-0.15))^2}{12}=\frac{169}{7500}\approx 0.0225333$ If you buy 200 stocks the corresponding values for the changes are
$\mathbb E(200X)=200\cdot \frac{0.37+(-0.15)}{2}=22$ and $Var(200X)=\frac{2704}{3}\approx 901.3333$
Now you can apply the central limit theorem in order to calculate that in $75$ days the value of all will not go above 1800 dollars. You have to assume that the change of the stocks at day $t$ does not affect the change in $t+1$ (much).