Mathematics 375 – Probability and Statistics 1 Problem Set 9, due: Thursday, 11/12

From Wackerly, Mendenhall, and Scheaffer: 5.11, 5.14, 5.15, 5.25, 5.26ab, 5.54, 5.59, 5.65, 5.76, 5.79, 5.80.

Additional Problem

A) As we proceed into Chapter 5, we will need to make increasingly heavy use of material on multiple integrals from MATH 241 (Multivariable Calculus). As a “warm-up” for this, and to fill in a proof for a fact we just stated in class, in this problem, we will develop a proof of the formula

B(p, q) = \frac{Γ(p)Γ(q)}{Γ(p + q)}

that we used in working with Beta-distributed random variables. (Thanks to Professor Anderson for suggesting this argument to derive the formula!!)

The idea is similar to what we did in class to derive the formula

∫_{−∞}^{∞} e^{−y^{2}/2}dy = \sqrt{2π}.

We will take two one-variable integrals, multiply them, treat as a double integral, do coordinate transformations, and derive the desired result.

The starting point will be the product:

(1)  Γ(p)Γ(q) = ∫_{0}^{∞} s^{p−1}e^{−s} ds · ∫_{0}^{∞} t^{q−1}e^{−t} dt

1) Show that the product on the right side in (1) is the same as the double integral

∫_{0}^{∞} ∫_{0}^{∞} s^{p−1}t^{q−1}e^{−(s+t)} dt ds = ∫ ∫_{R} s^{p−1}t^{q−1}e^{−(s+t)} dA

(where the region of integration R is the first quadrant in the (s, t)-plane).

2) To work with this, note that we can transform the integral by introducing a new coordinate system defined by u = s + t; s = s. This is a linear mapping from the (s, t) plane to the (u, s) plane defined by the matrix equation

( \begin{matrix}u \\ s \end{matrix} ) = ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ) ( \begin{matrix}s \\ t \end{matrix} )

The matrix has determinant of absolute value 1, hence defines an area-preserving mapping. The area element in the new coordinate system in this case is just dA = ds du. Show that this change of coordinates leads to a new equation

(2)  Γ(p)Γ(q) = ∫ ∫_{S} s^{p−1}(u − s)^{q−1}e^{−u} dA

where S is the image of R under the change of coordinate mapping.

3) Show that (2) can also be written as the following iterated integral:

(3)  Γ(p)Γ(q) = ∫_{0}^{∞} ∫_{0}^{u} s^{p−1}(u − s)^{q−1}e^{−u} ds du

4) Now, we will do a second change of coordinates, letting λ = s/u; u = u. This coordinate change is not area-preserving. The area element transforms to

ds du = u dλ du.

Assuming this (we will discuss the general change of variables theorem for multiple integrals in class if you have not seen it before), show that (3) becomes

(4)  Γ(p)Γ(q) = ∫_{0}^{∞} ∫_{0}^{1} λ^{p−1}(1 − λ)^{q−1}u^{p+q−1}e^{−u} dλ du

(Pay careful attention to the limits of integration on the “inside” integral. Where do they come from?)

5) Deduce the desired formula

B(p, q) = \frac{Γ(p)Γ(q)}{Γ(p + q)}

from (4). Very Cool!!!!