# The Central Limit Theorem

Suppose $X_1, \dots, X_{20}$ are i.i.d random variables with pdf $f(x) = 2x, 0 < x < 1$. Find $P(S < 10)$ where $S = X_1+ \cdots + X_{20}$.

So find $E(X)$ and $\text{Var}(X)$. Then $S$ has $N(20 \cdot E(X), 20 \cdot \text{Var}(X))$ distribution?

-
Are you sure that you should apply CLT here? – Ilya Mar 23 '12 at 14:44
@llya: Yeah I am applying it just as a very rough estimate. – jamies ross Mar 23 '12 at 14:48
@user22705: I know that. But the question asked to use the Central Limit Theorem. – jamies ross Mar 23 '12 at 14:51
ok, I see, do you know the statement of the Lindeberg Levy CLT? – Henrik Mar 23 '12 at 14:53
Why not apply the CLT here? $20$ is big enough to get a pretty close approximation to the normal distribution except with some pretty extreme cases, of which this is not one. – Michael Hardy Mar 23 '12 at 16:48

By CLT you have: $$\frac{S - 20\mu}{\sigma\sqrt{20}}\approx\mathcal N(0,1)$$ so $S\approx \mathcal N(20\mu,20\sigma^2)$ as you wrote in OP - so you're right.
Try the following $P(S<10)=P(S-20\mu<10-20\mu)=P(\frac{S-20\mu}{\sigma\sqrt{20}}<\frac{10-20\mu}{\sigma\sqrt{20}})\approx\Phi(\frac{10-20\mu}{\sigma\sqrt{20}})$
where $\phi(x)$ denotes the cumulative distribution function of a standard normal variable. All equalities above follows by equality of events. The last $\approx$ follows by the CLT.
Usually a capital $\Phi$ is used for the CDF and a lower-case $\varphi$ for the density, which is the derivative of the CDF. – Michael Hardy Mar 23 '12 at 16:49