Question

Suppose that contamination particle size (in micrometers) can be modeled as f(x) =

2x^−3

for x > 1. Determine the mean and standard deviation of X.

Answer 1

Mean: $E(X)=\int_{-\infty}^{+\infty}xf(x)dx=2\int_{1}^{+\infty}x^{-2}dx=-{\frac 2 x}|_1^{+\infty}=2$

Variance: $V(X)=E(X^2)-E^2(X)=\int_{-\infty}^{+\infty}x^2f(x)dx-4=2\int_{1}^{+\infty}x^{-1}dx-4=2ln(x)|_1^{+\infty}-4=+\infty-4=+\infty$

Standard deviation: $\sigma(X)=\sqrt{V(X)}=+\infty$