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January 16th, 2017, 12:31 PM  #1 
Newbie Joined: Jan 2017 From: Italy Posts: 10 Thanks: 0  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)
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: $\displaystyle P\left(T<11\right)=1P\left(T\geq11\right)=1P \left(\frac{T11.2}{\sqrt{0.1}}\geq\frac{1111.2}{\sqrt{0.1}}\right)=1\Phi\left(\frac{0.2}{\sqrt{0.1}}\right)=\Phi\left(\frac{0.2}{\sqrt {0.1}}\right)$ Is it right up to now? Now: for point b I really have no clue: what should I do? Thank you in advantage 
January 16th, 2017, 12:40 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,531 Thanks: 1390 
wat? The Normal distribution when specified as $N(x,y)$ almost always means $\mu = x$ $\sigma = y$ so $P[t < 11] = \Phi\left(\dfrac{1111.2}{0.1}\right) \approx 0.02275$ 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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central limit theorem, distribution, exercise, normal, normal distribution, standardized 
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