Calculate the mean, and the variance of the following probability density functions:
i) Uniform
ii) Rayleigh distribution
For the uniform density $p(x)=\frac{1}{2a}$ on $(-a,a)$ (zero elsewhere), symmetry gives $E[X]=\int_{-a}^{a}x\frac{1}{2a}\,dx=0$. Also $$ E[X^2]=\int_{-a}^{a}x^2\frac{1}{2a}\,dx=\frac{1}{2a}\cdot\frac{2a^3}{3}=\frac{a^2}{3}. $$ Hence $\operatorname{Var}(X)=E[X^2]-(E[X])^2=\frac{a^2}{3}$. For the Rayleigh density $p(x)=\dfrac{x}{a^2}e^{-x^2/(2a^2)}$ for $x\ge ____ ________ _____ _____ _________ ________ __________ ____ _______ _________ _________ _______.
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