Question

The proportion of people who respond to a certain mail-order solicitation is a random variable x having the following density function f(x)=2(x−1). Find the function g(x).

A) g(x)=2x−1
B) g(x)=x2−x+1
C) g(x)=x2−x
D) g(x)=x2−1

Answer 1

The function g(x) is found by integrating the given PDF f(x) from its lower limit, and the answer is g(x) = x^2 - x + 1. Option B is correct.

Step-by-step explanation:

The question is asking for the function g(x), which corresponds to the cumulative distribution function (CDF) of the given probability density function (PDF) f(x) = 2(x - 1). To find the CDF g(x), we need to integrate the PDF from the lower limit of x to the point of interest.

The proper steps to find g(x) involve the following:

Identify the lower limit of x in the PDF. In this case, since f(x) is a linear function starting from x = 1 (as the function becomes zero at x = 1), we consider the lower limit to be 1.

Integrate the function f(x) from 1 to x to get g(x).

By integrating f(x) = 2(x - 1) from 1 to x, we get:

g(x) = ∫(2(x - 1) dx) from 1 to x

g(x) = [x^2 - 2x + C] from 1 to x

g(x) = (x^2 - 2x + C) - (1^2 - 2(1) + C)

g(x) = x^2 - 2x + C - (1 - 2 + C)

g(x) = x^2 - 2x + 1

Because the constant C cancels out and we add 1 back due to evaluating from 1 to x, our final answer is g(x) = x^2 - x + 1, which corresponds to option B.