Question

In: Statistics and Probability

Data collected at an airport suggests that an exponential distribution with mean value 2.815 hours is...

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?

Solutions

Expert Solution

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


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