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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?

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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? –  user22705 Mar 23 '12 at 14:53
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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
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2 Answers 2

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.

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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.

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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
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