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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 \(c\) for each 尾-pdf are determined by the rule that the integral of a pdf over its valid range must equal 1. For \(f(x) = cx(1-x)^3\), \(f(x) = cx^4(1-x)^5\), and \(f(x) = cx^2(1-x)^8\), the respective values of \(c\) can be determined as the reciprocal of the Beta function \(B(a, b)\) with appropriate parameters \(a\), \(b\).

Step by step solution

01

Understanding Beta Distribution PDF

The pdf of the Beta distribution is \(f(x) = c x^{(a-1)} (1-x)^{(b-1)}\), where \(a\) and \(b\) are the shape parameters, \(x\) is the random variable, and \(c\) is the constant we need to find to ensure the integral of the function over its valid range equals 1. The Beta function \(B(a, b)\) is commonly used to represent this constant where \(B(a, b) = \int_0^1 x^{(a-1)} (1-x)^{(b-1)} dx\). Therefore, \(c\) can be found as the reciprocal of \(B(a, b)\).
02

Calculation for part (a)

Given \(f(x) = c x(1-x)^3\), which corresponds to \(a=2\) and \(b=4\). Using the formula for \(B(a, b)\), the constant \(c\) can be calculated as the reciprocal of the Beta function \(B(a, b)\).
03

Calculation for part (b)

Given \(f(x) = c x^4(1-x)^5\), which corresponds to \(a=5\) and \(b=6\). Similar to previous step, the constant \(c\) can be calculated as the reciprocal of the Beta function \(B(a, b)\).
04

Calculation for part (c)

Given \(f(x) = c x^2(1-x)^8\), which corresponds to \(a=3\) and \(b=9\). Same as the above steps, to find \(c\), compute the reciprocal of the Beta function \(B(a, b)\).

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