Suppose f(x) = 0.125x for 0 < x < 4. determine the mean and variance of x. round your answers to 3 decimal places.

Answers:

- mean: 2.667
- variance: 0.889

Explanation:

To get the mean and variance of x, we need to verify first whether...
- x is discrete or continuous random variable
- f is probability mass or probability density function

because if we cannot verify the 2 statements above, we can't compute the mean and the variance.

Since 0 < x < 4, x is a continuous random variable because x can be any positive number less than, which includes a non-integer.

Note that if the random variable is continuous and for any values of x in the domain of f, then f is a probability density function (PDF). 

Note that



Hence, for any x in the domain of f, 0 < f(x) < 1. Moreover, since x is a continuous random variable, thus f is a PDF. 

First, we use the following notations for mean and variance:

E[x] = mean of x
Var[x] = variance of x

Since f is a probability density function, we can use the following formulas for the mean and the variance of x:





To compute for the mean of x,



The integral seems complicated because of the infinity sign. But because the domain of f is the set of positive numbers less than 4, that is,



the bounds of the integral for the mean can be changed from  to  so that 



Hence, the mean is computed as 



Since the formula for variance is computed as 



we must first compute for  for which



Similar to the computation of integral of the mean, we take note that 



so that we can change the bounds of the integral, that is,



Hence,



Because ,