The annual rainfall (in inches) in a certain region is

snijelihd 2021-11-11 Answered
The annual rainfall (in inches) in a certain region is normally distributed with \(\displaystyle\mu={40}\) and \(\displaystyle\sigma={4}\)
What is the probability that, starting with this year, it will take over 10 years before a year occurs having a rainfall of over 50 inches? What assumptions are you making?

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Expert Answer

Michele Tipton
Answered 2021-11-12 Author has 1900 answers
Given that the annual rainfall is normally distributed with mean \(\displaystyle\mu={40}\) inches and standart deviation \(\displaystyle\sigma={4}\) inches. Let X be the normal random variable of the annual rainfall.
To find: P
Solution: We will first find the probability of rainfall above 50 inches in one year.
\(\displaystyle{P}{\left[{X}{>}{50}\right]}={1}-{P}{\left[{x}\leq{50}\right]}\)
\(\displaystyle={1}-{P}{\left[{\frac{{{X}-\mu}}{{\sigma}}}\leq{\frac{{{50}-{40}}}{{{4}}}}\right]}\)
\(\displaystyle={1}-{P}{\left[{Z}\leq{\frac{{{5}}}{{{2}}}}\right]}\)
\(\displaystyle={1}-\Phi{\left({\frac{{{5}}}{{{2}}}}\right)}\)
\(\displaystyle{1}-{0.9938}\)
\(\displaystyle={0.0062}\)
P[non-occurence of rainfall above 50 inches]=1-0.0062=0.9938
Assuming independence over the years, we get
P[It will take over 10 years or more before an year with a rainfall above 50 inches]\(\displaystyle={\left({0.9938}\right)}^{{{10}}}\)
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