Data collected at Toronto Pearson International Airport suggests that an exponential distribution with mean value 2.725 hours is a good model for rainfall duration (Urban Stormwater Management Planning with Analytical Probabilistic Models,
2000, p. 69).
a. What is the probability that the duration of a particular
rainfall event at this location is at least 2 hours? At most
3 hours? Between 2 and 3 hours?
b.What is the probability that rainfall duration exceeds the
mean value by more than 2 standard deviations? What is
the probability that it is less than the mean value by more
than one standard deviation?
The question in bold is the part I am having difficulty with. Here is what I tried:
$P(X < \mu\text{ by more than one }\sigma) = 1 – P(X > \mu + \sigma) = 1 – F(\mu + \sigma; 0.36697)$
This doesn't seem correct, though. Could someone possibly help me?
Best Answer
The question does not refer to the event that $X$ is less than an amount that is one standard deviation more than the mean. Rather, it asks about the event that $X$ is less than the mean by more than one standard deviation.
Look at this phrase: $$ X < \mu\text{ by more than one }\sigma. $$ That's the same as $$ X< \mu-\sigma. $$ That is not the same as $$ X\not>\mu+\sigma. $$ Remember what the standard deviation of an exponential distribution is, and go on from there.