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January 16th, 2017, 12:31 PM   #1
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Exercise with Normal Distribution

Hello everybody, I need your help in order to solve this exercise:

One athlete timing, $\displaystyle T$ (in seconds) on 100m discipline is a random variable $\displaystyle T \sim N(11.2,0.1)$.
Each time the athlete reaches a timing better than 11 seconds, it wins 500$. During next year he will play 35 matches wich are one indipendent from the others (i.e. there are enough days between races to recover)
  1. Compute the probability that the athlete will perform a timing better than 11 seconds.
  2. Approximatively compute the probability that during the year he will win at least 8000$.

Regarding point a I made $\displaystyle T$ a Normal Standardized in this way:

$\displaystyle \frac{T - 11.2}{\sqrt{0.1}} \sim N \left(0,1 \right)$
then, through central limit theorem:
$\displaystyle P\left(T<11\right)=1-P\left(T\geq11\right)=1-P \left(\frac{T-11.2}{\sqrt{0.1}}\geq\frac{11-11.2}{\sqrt{0.1}}\right)=1-\Phi\left(\frac{-0.2}{\sqrt{0.1}}\right)=\Phi\left(\frac{0.2}{\sqrt {0.1}}\right)$

Is it right up to now?
Now: for point b I really have no clue: what should I do?

Thank you in advantage
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January 16th, 2017, 12:40 PM   #2
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The Normal distribution when specified as $N(x,y)$ almost always means

$\mu = x$

$\sigma = y$


$P[t < 11] = \Phi\left(\dfrac{11-11.2}{0.1}\right) \approx 0.02275$

Now just use that probability as the $p$ in a Binomial(35,p) distribution to find the probability he/she wins at least 16 times, i.e. \$8000.
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