Question

Data collected at Toronto Pearson International Airport suggest that an exponential distribution with mean value 2,725 hours is a good model for rainfall duration. What is the probability that the duration of a particular rainfall event at this location is at least 2 hours?

Answer 1

Answer:

$P(X >2)$

And we can calculate this with the complement rule like this:

$P(X>2) = 1-P(X<2)$

And using the cdf we got:

$P(X>2) = 1- [1- e^{-\lambda x}] = e^{-\lambda x} = e^{-\frac{1}{2.725} *2}= 0.480$

Step-by-step explanation:

Previous concepts

The exponential distribution is "the probability distribution of the time between events in a Poisson process (a process in which events occur continuously and independently at a constant average rate). It is a particular case of the gamma distribution". The probability density function is given by:

$P(X=x)=\lambda e^{-\lambda x}, x>0$

And 0 for other case. Let X the random variable of interest:

$X \sim Exp(\lambda=\frac{1}{2.725})$

Solution to the problem

We want to calculate this probability:

$P(X >2)$

And we can calculate this with the complement rule like this:

$P(X>2) = 1-P(X<2)$

And using the cdf we got:

$P(X>2) = 1- [1- e^{-\lambda x}] = e^{-\lambda x} = e^{-\frac{1}{2.725} *2}= 0.480$