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 ? = 0.01327.
(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? (Round your answer to
two decimal places.)
The concept of the probability and the exponential distribution are used to solve this problem.
Integrate the exponential function within the provided limits to determine the required probability.
An exponential distribution is a continuous distribution.
A continuous random variable X is said to have an exponential distribution with a parameter , if its probability distribution function is given by,
Here. is the exponential distribution parameter.
The mean and standard deviation of the exponential distribution is given as,
Here, is the mean of the function and is the standard deviation.
(a.1)
Obtain the exponential distribution parameter from the problem.
Write the probability distribution function.
Integrate above function with respect to distance and take limits from to .
(a.2)
Write the probability distribution function.
Integrate the above function with respect to distance and take limits from to .
(a.3)
Write the probability distribution function.
Integrate the above function with respect to distance and take limits to .
(b)
Write the expression to calculate the mean.
Substitute for .
Since the expression of the mean and standard deviation of the exponential distribution are same. Therefore,
Calculate the probability that distance exceeding the mean by more than two standard deviations.
Further solve as,
(c)
Calculate the value of median distance.
Take on both sides and solve.
Substitute for .
Ans: Part a.1
The probability of the distance at most is .
Part a.2The probability of distance at most is .
Part a.3The probability of distance lies between to is .
Part bThe probability of the distance exceeding the mean by more than two standard deviations is .
Part bThe value of the median distance is .
Let X denote the distance (m) that an animal moves from its birth site tothe...
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