Question

I. Derive the mean and variance of a uniformly distributed random variable X where X ~ U (a, b). 

II. Let X be uniformly distributed in -2 ≤ x ≤ 2 Find:

a) P(X < 1)

b) P( X - 1 ≥ 1/2 ).

Answer 1

I. $\mu=E(X)=\int_a^b\frac{x}{b-a}dx=\frac{1}{2}\frac{b^2-a^2}{b-a}=\frac{a+b}{2}.$

$Var(X)=\int_a^b\frac{x^2}{b-a}dx-\mu^2=\frac{(b^3-a^3)}{3(b-a)}-\frac{(a+b)^2}{4}=\frac{(b-a)^2}{12}.$

II.

a) $P(X<1)=\frac{1-(-2)}{2-(-2)}=\frac{3}{4}=0.75.$

b) $P(X-1\ge\frac{1}{2})=P(X\ge\frac{3}{2})=\frac{2-\frac{3}{2}}{2-(-2)}=\frac{\frac{1}{2}}{4}=\frac{1}{8}=0.125.$