Let X denote the distance (m) that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that for banner-tailed kangaroo rats, X has an exponential distribution with parameter λ = 0.01342. (a) What is the probability that the distance is at most 100 m? At most 200 m? Between 100 and 200 m? (Round your answers to four decimal places.) at most 100 m at most 200 m between 100 and 200 m (b) What is the probability that distance exceeds the mean distance by more than 2 standard deviations? (Round your answer to four decimal places.) (c) What is the value of the median distance? Hint: Find a such that P(X≤a)= 0.50 (Round your answer to two decimal places.) m (d) Only 5% of animals will move farther than what distance? Hint: Find a such that P(X≤a)= 0.95 . (Round your answer to two decimal places.) m

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:

[tex]P(X=x)=\lambda e^{-\lambda x}[/tex]

Solution to the problem

For this case we have that X is represented by the following distribution:

[tex] X\sim Exp (\lambda=0.01342)[/tex]

Is important to remember that th cumulative distribution for X is given by:

[tex] F(X) =P(X \leq x) = 1-e^{-\lambda x}[/tex]

Part a

For this case we want this probability:

[tex] P(X \leq 100)[/tex]

And using the cumulative distribution function we have this:

[tex] P(X \leq 100) = 1- e^{-0.01342*100} =0.7387[/tex]

[tex] P(X \leq 200) = 1- e^{-0.01342*200} =0.9317[/tex]

[tex] P(100\leq X \leq 200) = [1- e^{-0.01342*200}]-[1- e^{-0.01342*100}] =0.1930[/tex]

Part b

Since we want the probability that the man exceeds the mean by more than 2 deviations

For this case the mean is given by:

[tex] \mu = \frac{1}{\lambda}=\frac{1}{0.01342}= 74.516[/tex]

And by properties the deviation is the same value [tex] \sigma = 74.516[/tex]

So then 2 deviations correspond to 2*74.516=149.03

And the want this probability:

[tex] P(X > 74.516+149.03) = P(X>223.547)[/tex]

And we can find this probability using the complement rule:

[tex] P(X>223.547) = 1-P(X\leq 223.547) = 1-[1- e^{-0.01342*223.547}]=0.0498[/tex]

Part c

For the median we need to find a value of m such that:

[tex] P(X \leq m) = 0.5[/tex]

If we use the cumulative distribution function we got:

[tex] 1-e^{-0.01342 m} =0.5[/tex]

And if we solve for m we got this:

[tex] 0.5 = e^{-0.01342 m}[/tex]

If we apply natural log on both sides we got:

[tex] ln(0.5) = -0.01342 m[/tex]

[tex] m = \frac{ln(0.5)}{-0.01342}=51.65[/tex]

Part d

For this case we have this equation:

[tex] P(X\leq a) = 0.95[/tex]

If we apply the cumulative distribution function we got:

[tex] 1-e^{-0.01342*a} =0.95[/tex]

If w solve for a we can do this:

[tex] 0.05= e^{-0.01342 a}[/tex]

Using natural log on btoh sides we got:

[tex] ln(0.05) = -0.01342 a[/tex]

[tex] a = \frac{ln(0.05)}{-0.01342}=223.23[/tex]