In: Statistics and Probability
Data collected at an airport suggests that an exponential distribution with mean value 2.815 hours is a good model for rainfall duration.
(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?
(Round your answers to four decimal places.)
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 4 standard deviations? (Round your answer to four decimal places.)
What is the probability that it is less than the mean value by more than one standard deviation?
Solution:
Given:Mean=1/=2.815 hours
Let X be the random variable that denotes the rainfall duration and it follows exponential distribution.
The CDF of exponential distribution is
F(x;)=P(Xx)=1-e-1/
a) Probability that the duration of a particular rainfall event at this location is at least 2 hours is calculated as follows:
P(X2)=1-P(X<2)=1-(1-e-2/2.815)=0.4914
Probability that the duration of a particular rainfall event at this location is at most 3 hours is calculated as follows:
P(X3)=1-e-3/2.815=1-0.3445=0.6555
Probability that the duration of a particular rainfall event at this location is between 2 and 3 hours is calculated as follows:
P(2X3)=P(X3)-P(X2)=(1-e-3/2.815)-(1-e-2/2.815)=e-2/2.815-e-3/2.815=0.4914-0.6555=0.1641
b) Probability that rainfall duration exceeds the mean value by more than 4 standard deviations is as follows:
P(X>)=P(X>3)=e-3/=e-3=0.0498
c)Probability that it is less than the mean value by more than one standard deviation is as follows:
P(X<)=P(X)=P(X2)=1-e-2/ =e-2=0.1353