Problem based on distributions

Question

Probability Background

Prove that the mean E(x), the expectation E(x2) and the variance σ2x by using

a- PDF p(x), where

E(x) = _-∞∫^∞ x p(x) dx

E(x2) = _-∞∫^∞ x² p(x) dx

-∞

σ²_x = E(x²) - (E(x))² .

b- Characteristics function, where

ψ(jv) = _-∞∫^∞ ejvx p(x) dx

E(x) = -j dψ(jv)/dv|_v=0

E(x)² = (-j)²d²ψ(jv)/dv²|_v=0

σ²x = E(x²) - (E(x))²

for the following distributions:

1- Uniform Distribution

2- Chi-Square Distribution

3- Rayleigh Distribution

4- Rician distribution

Attachment:- Probability Background.rar