Question

Trainees must complete a specific task in less than 2 minutes. Consider the probability density function f(x) = 1 - 0.5x for 0 < x < 2. Find the cumulative distribution function F(x).

Answer 1

The cumulative distribution function (CDF) for the given probability density function is F(x) = x - 0.25x^2.

Step-by-step explanation:

The cumulative distribution function (CDF), denoted as F(x), gives the probability that a random variable is less than or equal to a particular value, x. In this case, the probability density function (PDF) is given as f(x) = 1 - 0.5x for 0 < x < 2. To find the cumulative distribution function, we need to integrate the PDF from 0 to x.

So:

F(x) = ∫(0 to x) [1 - 0.5t] dt

Using the integral, we get:

F(x) = [t - 0.25t^2] evaluated from 0 to x

Substituting in the limits and simplifying, we have:

F(x) = x - 0.25x^2