(1) We have spent a lot of time considering transformations of random variables.
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(1) We have spent a lot of time considering transformations of random variables. In many cases, simple transformations of random variables (like shiftin or ig) produce other random variables from the same family. For example, if you shift a uniform, you get a (different) uniform. Similarly, if you scale a uniform, you get a (different) uniform. Shifting usually keeps a distribution in the same family, but not always. A critical counterexample would be a distribution that was required to fall in a certain domain, but shifting it violates that domain. Give an example of a distribution that stays within its family when shifted, and another one that changes to something different when shifted. Do not cite the uniform. a. b. Scaling almost always keeps a distribution in the same family. The simple engineering explanation for this is that scaling is the same as changing units, and changing units shouldn't change anything fundamental about the quantity being measured (an important exception would be a distribution intended to model dimensionless quantities, like the beta distribution). Give an example (other than the uniform) of a distribution that remains in the same family after scaling. (2) In the previous homework, we started with X-uniform(0,1), and used the transformation Y =-In X to produce exponential (A) random variables. We convinced ourselves this was correct because the histogram seemed to look like an exponential pdf. Use mathematics to show that this is the correct result (3) In the previous homework, you used histograms to show what would happen if you added together three independent exponential random variables. Repeat that experiment for 5, 10, and 15 independent exponentials. Comment on the shapes.Explanation / Answer
a. Normal distribution does not change when shifted whereas alpha or beta distribution changes.
b. Exponential distribution does not change after changing its scale.
You may draw graphs to understand these even better.
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