Continuous probability finding mean
Hi, I’m attempting part three of this question but I m not sure how to do it, I get something along the lines of xarctanh(x) - log(cosh(x)) as the indefinite Integral ( minus some of the s and mu constants) but this seems to evaluate to infinity and minus I finite at the limits?
Is there a way to do this without integration? I have no access to solutions but examiners comments mention that a lot of people used I refraction by parts which wasn’t really the way it was supposed to be done
Is there a way to do this without integration? I have no access to solutions but examiners comments mention that a lot of people used I refraction by parts which wasn’t really the way it was supposed to be done
Original post by DFranklin
Surely you can just argue symmetry about x=mu?
I guess, would that be sufficient for a 5 mark question? And does P( -inf < x < mu) = 1/2 imply that mu is the mean?
Original post by grhas98
I guess, would that be sufficient for a 5 mark question? And does P( -inf < x < mu) = 1/2 imply that mu is the mean?
No. This implies that mu is the median.
To prove mu is the mean you need to show that .
(edited 1 year ago)
Original post by RDKGames
No. This implies that mu is the median.
To prove mu is the mean you need to show that .
To prove mu is the mean you need to show that .
I know but the problem comes when trying to integrate I end up with a function that seems to evaluate to infinity + - infinity I don’t really know what to do with it
Original post by grhas98
I know but the problem comes when trying to integrate I end up with a function that seems to evaluate to infinity + - infinity I don’t really know what to do with it
Doing a simple substitution so something like z = x-m/2s should bring the mu outside the sech^2, then its just a case of using the fact that the integral of z*sech^2 will be zero as its odd and the integral of m*sech^2 will be m as the sech^2 is a pdf (normalized to unity).
(edited 1 year ago)
Original post by grhas98
I guess, would that be sufficient for a 5 mark question? And does P( -inf < x < mu) = 1/2 imply that mu is the mean?
As RDK says, your last statement is about the median, not the mean.
To my mind, using symmetry to find the mean is sufficient as long as you also justify that the (expectation) integral converges (c.f. the Cauchy distribution 1/(pi(1+x^2)) which is symmetrical about x = 0 but whose mean is actually undefined).
If you want to, you can easily formalise the symmetry argument - it's essentially just the observation that if f is an even function, and a bit of manipulation / substitution.
(edited 1 year ago)
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