Final answer:
To find the distribution function of the random variable X, we need to find the cumulative distribution function (CDF). The CDF can be found by integrating the probability density function (PDF) in two parts.
Step-by-step explanation:
The probability distribution function (PDF) is given by: f(x) = x for 0 < x < 1, 2 - x for 1 ≤ x < 2, and 0 elsewhere. To find the cumulative distribution function (CDF), we need to integrate the PDF from negative infinity to x. Let's break it down into two parts: 0 < x < 1 and 1 ≤ x < 2.
For 0 < x < 1, the CDF can be found by integrating x with respect to x from 0 to x. The integral of x is (x^2)/2. So, the CDF for 0 < x < 1 is (x^2)/2. For 1 ≤ x < 2, the CDF can be found by integrating 2 - x with respect to x from 1 to x. The integral of 2 - x is 2x - (x^2)/2 - 1. So, the CDF for 1 ≤ x < 2 is 2x - (x^2)/2 - 1.
To find the overall CDF, we can piece together these two parts. For 0 < x < 1, the CDF is (x^2)/2, and for 1 ≤ x < 2, the CDF is 2x - (x^2)/2 - 1. So, the CDF for the entire range of x is: For 0 < x < 1: CDF = (x^2)/2. For 1 ≤ x < 2: CDF = 2x - (x^2)/2 - 1