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Chapter 3: Problem 19

Determine the constant \(c\) in each of the following so that each \(f(x)\) is a \(\beta\) pdf: (a) \(f(x)=c x(1-x)^{3}, 0

Short Answer

Expert verified
The constants for the beta PDFs are: (a) \(c = 30\), (b) \(c = 462\), and (c) \(c = 1287\).

Step by step solution

01

Setting up the integral for (a)

In each case, the constant \(c\) is whatever value makes the integral of the function over its range equal to one. It means that for (a), the equation is \( \int_{0}^{1}c x(1-x)^{3} dx = 1\).
02

Evaluate the integral for (a)

This can be evaluated by techniques of integration: \( \int_{0}^{1} x(1-x)^3 dx = \left[-\frac{(1-x)^4x}{5}\right]_0^1 - \frac{4}{5}\int_{0}^{1}(1-x)^3 dx\). This gives \(\frac{4}{5}\left[\frac{1}{4}\right - \frac{1}{5}\int_{0}^{1}(1-x)^2 dx\]. Continuing using the same method, we find that it equals \(\frac{1}{30}\). So for \(f(x) = c x(1-x)^3\) to be a beta pdf, \(c = 30\).
03

Setting up and evaluating the integral for (b)

Same idea for (b). We solve it to find that \(c = 462\).
04

Setting up and evaluating the integral for (c)

For (c), we again use same process to find that \(c = 1287\).

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