Introduction to Probability with Statistical Applications by Géza Schay

Author:Géza Schay
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(6.29)

For continuous X, the general proof is beyond the scope of this book and is therefore omitted3. However, if g is one-to-one and differentiable, then the proof is easy, and goes like this:

By Definition 6.1.1,

(6.30)

where f Y (y) is given by the methods of Section 5.​3 as

(6.31)

Thus, changing variables in Equation 6.30 from to we get

(6.32)

as stated in Equation 6.24. ■

Example 6.1.10.

Average Area of Circles.

Assume that we draw a circle with a random radius R, uniformly distributed between 0 and some constant a. What is the expected value of the area Y = π R 2 of such a circle?

Now,

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