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Chapter 1: Problem 12

Find the cdf \(F(x)\) associated with each of the following probability density functions. Sketch the graphs of \(f(x)\) and \(F(x)\). (a) \(f(x)=3(1-x)^{2}, 0

Short Answer

Expert verified
(a) CDF: \(F(x) = 1 - (1-x)^3, median 鈮� 0.793, 25th percentile 鈮� 0.63\), (b) CDF: \(F(x) = 1 - 1/x, median = 2, 25th percentile 鈮� 1.33\), (c) CDF: For \(0 < x < 1, F(x) = x/3\), For \(2 < x < 4, F(x) = x/3 - 2/3\), median not defined, 25th percentile not defined.

Step by step solution

01

Calculate the Cumulative Distribution Function (CDF)

The cdf is the integral of the pdf from negative infinity to x. (a) \(F(x) = \int_0^x 3(1-t)^2 dt = [ - (1-t)^3]_0^x = 1 - (1-x)^3\) for \(0 < x < 1\) (b) \(F(x) = \int_1^x 1/t^2 dt = [- 1 / t]_1^x = 1 - 1/x \) for \(1 < x < \infty\)(c) \(F(x) = \int_0^x 1/3 dt = [t/3]_0^x = x/3\) for \(0 < x < 1\) and \(F(x) = \int_2^x 1/3 dt = [t/3]_2^x = x/3 - 2/3\) for \(2 < x < 4\)
02

Sketch the Graphs of f(x) and F(x)

Sketching the graphs may demonstrate how the pdf and cdf relate to each other. The x-axis on a graph can be used to depict the value of x, the y-axis on the left to display the value of \(f(x)\), the pdf, and the y-axis on the right to represent the value of \(F(x)\), the cdf. In each section of the graph where \(f(x) = 0\), \(F(x)\) will be a horizontal line, and in each section where \(f(x) 鈮� 0\), \(F(x)\) will be an increasing function.
03

Calculate Median and 25th Percentile

The median and the 25th percentile \(q\) of the distribution can be computed by setting \(F(q) = 0.25\) and \(F(m) = 0.5\) respectively and then solving for \(q\) and \(m\). (a) Median: \(1 - (1-m)^3 = 0.5\) leads to \(m 鈮� 0.793\). 25th percentile: \(1 - (1-q)^3 = 0.25\) leads to \(q 鈮� 0.63\).(b) Median: \(1 - 1/m = 0.5\) leads to \(m = 2\). 25th percentile: \(1 - 1/q = 0.25\) leads to \(q 鈮� 1.33\).(c) Median: \(x/3 = 0.5\) for \(x < 1\), and \(x/3 - 2/3 = 0.5\) for \(2 < x < 4\). However, no solutions exist in the defined range so no median can be computed. Similarly, for 25th percentile, the equation \(x/3 = 0.25\) for \(x < 1\), and \(x/3 - 2/3 = 0.25\) for \(2 < x < 4\) doesn't have solutions.

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