1. In many cases, simple transformations of random variables (like shifting or s

Question

1. In many cases, simple transformations of random variables (like shifting or scaling) 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.

a. 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 Do not cite the uniform.

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.

Explanation / Answer

(a) Let us say X represents a normal distribution with mean M and standard deviation 1

Then we know Z = (X-M) is still normal with mean 0 and standard deviation. Thus even after shifting family remains the same.

Now, let us say a pdf is like this exists

X = Cx^2 0<X<1

0 otherwise

if you shift it let us say Y = X + 5, then Y has a different distribution that X. Thus, it changes to something different

(b) Let us again consider the X which again follows the normal distribution with mean 0 and standard deviation S

Then, we know Y = X/S follows a standard normal distribution with mean 0 and standard deviation 1. Thus, it belongs to the same family even after rescaling.

Related Questions

Navigate

• Browse (All)
• Browse 1
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.