I have some problems with this
The annual rainfall (in inches) in a certain region is normally distributed with μ = 40 and σ = 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?
Let X be a normal random variable with μ = 40 and σ = 4 that represents the annual rainfall. $$P(X>= 50)=P((X-\mu )/\sigma \ge (50-10)/4 =2,5)=1-\Phi (2,5)=0.9938$$
Let Y be a geometric random variable with parameter $$p=0,9938$$ The probability that it will take over 10 years before a year occurs having a rainfall of over 50 inches would be $$P(Y=11)=0,9938^{10}*(1-0,9938)$$ because at the eleventh year we will have surely a rainfall of over 50 inches. But the solution on the book is $$0,9938^{10}$$ only.
Why?
I'm assuming that each event is independent from the others.
$\endgroup$ 11 Answer
$\begingroup$It says over ten years, not exactly ten years. So that's equivalent to the first 10 years having less than 50 inches.
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