by scale effects do you mean you want an explanation of why
and
?
Suppose X has a probability density given by:
1) Find the probability density of
2) Find the mean and variance of
Could someone please explain or refer me to someone website which explains the 'scale effects'. I tried reading it in my textbook, but don't really understand it.
Ok, lets examine the CDF of Y.
Step 1:
Step 2:
Now, to get any pdf from a cdf, we just differenciate.
You can do this derivative easily using the chain rule
Notation point: means the derivative of F with respect to x, evaluated at the point
But by definition, the derivative of F is the pdf (ie, F' = f).
To explain why F(X) = G(aX + b) you can follow the argument in step 1 again, but try to find F() instead of G(). I've done it as a spoiler in case you get stuck
Spoiler:
Im not clear on what you mean, there are 2 ways to get the pdf of Y.
Short way:
just plug the numbers into the formula we just derived
Long way:
Step 1:
Find the CDF of X
HintYou can either integrate, or notice that we are dealing with the pdf of an exponential distribution. You can lookup the CDF of an exponential distribution
Step 2:
Find the CDF of Y (use the same reasoning i did in the derivation above)
Step 3:
Differentiate the CDF of Y to get the pdf of Y.