Question

Let X have a continuous distribution with pdf f where the graph of f on [0,1] consists of the following: the line segment joining the origin with the point (1/4,4/3), the line segment joining the points (1/4,4/3) and (3/4,4/3), and the line segment joining the points (3/4,4/3) and (1,0). For all x outside the interval (0,1),f(x)=0. Obtain the cumulative distribution function (cdf) F of X. Sketch the graphs of f and F.

Answer 1

The cumulative distribution function (CDF) of X can be obtained by integrating the probability density function (pdf) and finding the area under the curve between different intervals. The pdf is defined by different equations for different intervals on the range [0,1]. The graphs of the pdf and cdf consist of line segments connected at their endpoints.

Step-by-step explanation:

The cumulative distribution function (CDF) of X can be obtained by finding the area under the probability density function (pdf) between 0 and x. Let's break down the process step by step:

1. First, we need to find the equation for the pdf. Given the description, the equation for the pdf is as follows: f(x) = 4/3 for 0 ≤ x ≤ 1/4, f(x) = -2x + 4/3 for 1/4 ≤ x ≤ 3/4, and f(x) = -4x + 4 for 3/4 ≤ x ≤ 1.
2. Next, we integrate the pdf equation to obtain the cdf equation. For 0 ≤ x ≤ 1/4, the cdf is F(x) = (4/3)*x. For 1/4 ≤ x ≤ 3/4, the cdf is F(x) = -(x^2 - 4x/3 + 1/16). For 3/4 ≤ x ≤ 1, the cdf is F(x) = -(x^2 - 4x/3 + 13/16).
3. Finally, we need to plot the graphs of the pdf and cdf. The pdf will consist of three line segments as described, while the cdf will be a continuous curve connecting the endpoints of the line segments.