Statistics Question

Question

A professor must spend different amounts of time with each student to help with various topics during office hours. These times follow an exponential distribution with the average amount of time of μ=21

 minutes.

a) Find m

 

m=  

b) Using your answer from part a), write the probability density function.

f(x)=

 

Answer 1

In statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process. Now generally an exponential distribution is defined using a rate parameter of lambda (λ) which would be number of events per unit of time. Given the average or mean (mu or µ) for an exponential distribution, lambda follows the following equation:

λ = 1 ⁄ µ or λ = 0.047619 events per minute (remember, lambda is a rate)

With this information we can write the PDF of the distribution. If we accept the standard definition , which is derived from a Poisson point process without further proof, as being

f(x,λ) = λe^-λx for x ≥ 0

(we are ignoring negative times because the result is 0)

then for the given problem conditions we generate the following probability density function( for x in minutes):

.0476190e-^0.0476190x

So much for the definitions and some of the theory behind the exponential function. Now to the student's questions.

Find the m where m = ?

Part a) Not possible with the information given as m is neither defined nor intuitively determinable from standard statistical formulations. I considered m might be mean, but it could also be minutes. m being the mean makes no sense as the mean is given in the question. minutes also makes no sense as the parameterization uses lambda in this case is events per minute. So, without additional information, part a can not be answered.

Part b) Using your answer from part a), write the probability density function f(x)= ?

Also not possible as the probability density function for an exponential distribution is defined for two parameters, lambda and x, not just x. There is NO f(x) that can be a probability distribution function for the exponential distribution. If we make the wild assumption that the undefined m is actually λ there could be an f(x, λ=.m or 0.0476190) - which is given above, but no f(x)

For the future, please make sure that your questions make sense before posting. If you don't know enough to properly write the question it is doubtful that a proper answer would be meaningful to you. Good luck.