Determine average coordinate and the fluctuation

Question

Discussion:

1. A particle is distributed along the positive x axis with a probability density

P(x) = C x² exp(-(x/a)²). Determine it's average coordinate , and the fluctuation <Δx²> ; where <Δx>= x-

2. The coordinate of a particle is distributed along the WHOLE x-axis with a probability density

P(x) = C /(x² +a²)^3/2.

Determine C, , and the fluctuation <Δx²>