# CLT problem.

• Dec 11th 2011, 10:39 AM
juanma101285
CLT problem.
Hi,

I have the following problem that comes with the solution, but I don't understand where $\displaystyle X/120 \approx N(6, 1/4)$ comes from. Thanks a lot for your help! :)

The problem and the solution are here:

Problem:
I roll a fair die until I get a six. I do this 120 times. Estimate the probability that I need more than an average of 6.5 rolls each time.

Solution:
Let $\displaystyle X$ be the total number of turns needed. Then $\displaystyle X \backsim NB(120,1/6)$ is the sum of $\displaystyle 120$ independent RVs with distribution $\displaystyle Geom(1/6)$. The CLT gives $\displaystyle X/120 \approx N(6,1/4)$, so $\displaystyle p(X/120>6.5)=1-p((X/120-6)/(1/2)<1) \approx 1 - \Phi (1)= \Phi(-1)$.
• Dec 12th 2011, 08:39 AM
emakarov
Re: CLT problem.
Quote:

Originally Posted by juanma101285
I have the following problem that comes with the solution, but I don't understand where $\displaystyle X/120 \approx N(6, 1/4)$ comes from.

For Geom(p), the mean $\displaystyle \mu=1/p$ and the variance $\displaystyle \sigma^2=\frac{1-p}{p^2}$ (Wikipedia). For $\displaystyle p = 1/6$, we have $\displaystyle \mu=6$ and $\displaystyle \sigma^2=30$. By CLP, $\displaystyle X/120\sim N(\mu,\sigma^2/120)$, i.e., $\displaystyle X/120\sim N(6,1/4)$.