Question

A random variable X has probability density function f (x) = cx for 0 < x < 1/2 and f (x) = 0 otherwise, where c is a constant.

Find the value for the constant c that properly defines the probability density function.

Answer 1

The value for the constant c that defines the probability density function f(x) = cx for 0 < x < 1/2 and f(x) = 0 otherwise, ensuring that the total area under the curve is one, is c = 8.

Step-by-step explanation:

To find the value for the constant c that properly defines the probability density function (pdf) f(x) = cx for 0 < x < 1/2 and f(x) = 0 otherwise, we must ensure that the total area under the pdf is equal to 1. This is one of the fundamental properties of a pdf in continuous probability functions.

Since the function is non-zero only between 0 and 1/2, we can find the area under the curve f(x) by integrating the function from 0 to 1/2. The required integral would be ∫ cxdx evaluated from 0 to 1/2.

Performing this integration, we get the area is equal to (c/2) * (1/2)^2, which simplifies to c/8. To satisfy the condition that the total area under the curve is one, we set c/8 = 1, and solving for c gives us the value c = 8.