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Transformation of Variables in Probability Distribution

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I've struggled for 3 days to come up with something approaching a relevant answer but am now desperate.

Could you solve Q3, both a) and b) parts from the Exercise Sheet attached?

Happy to pay 2 credits for both answers.

Thank you very much.

3. The random variable X has an exponential distribution with mean µ. Let Y = e
(a) Use the method of transformations to find the probability density function of
Y . [ 4 marks ]
(b) Use the method of distribution functions to find the cumulative distribution
function of Y .

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Solution Summary

This solution explains in small details how to transform a probability distribution from one random variable to another

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Here is a somewhat modified version of your presentation for the method of transformations.
I like it done in this way because I think it makes all the details transparent.

First, we write down the exponential distribution:

dP=1/μ e^(-X/μ) dX. X∈[0,∞) (1)

It may be that you are more used to start from the PDF

p(x)=dP(x)/dx=1/μ e^(-X/μ), (2)

however it is often ...

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