Suppose that contamination particle size (in micrometers) can be modeled as can be modeled as $$f (x) = 2x-^3 for 1 < x. Determine the mean of X. What can you conclude about the variance of X?

Mean = 2

What i can conclude about the variance is that it doesn't exist.

Step-by-step explanation:

We want to determine the mean of "X".

So first of all, let the probability density function (f) of the random variable X be;

f (x) = 2x^ (-3)

This can be simply written as;

f (x) = 2/x³, x > 1

Thus;

Mean E (X) = (∞,1∫) 2/x³ (xdx)

= (∞,1∫) (2/x²) dx

Integrating, we have;

E (X) = - 2/x at (∞,1)

Thus E (X) = (-2/∞) - ( - 2/1)

= 0 + 2 = 2.

So mean E (X) = 2

Variance E (X²) = = (∞,1∫) 2/x³ (x²dx)

= (∞,1∫) (2/x) dx

Integrating, we find that x becomes 1 and thus there's no way to apply the boundary (∞,1). Thus, the integral can be said to be diverging and thus doesn't exist.

Since the integral doesn't exist, the variance doesn't also exist.