Uniform (Continuous)

\[ f_X(x) = \frac{1}{a}; \qquad a \le x \le 1 \]

Linear Transformation of Uniform(0,1) : A time-saving property of the uniform random variable is all uniform random variable can be expressed as a linear transformation of the \(U \sim \text{Uniform}(0,1)\) with PDF,

\[ f_U(u) = 1; \qquad 0 \le u \le 1 \]

Variance : Since the second moment is,

$$
E(U^2) = \int_0^1 x^2 ~du = \frac{1}{3}
$$

$$
\text{Var}(U) = \frac{1}{12}
$$

CDF : $\( F_U(x) = \frac{x-a}{b-a} \)$

Expectation : $\( E(X) = a + (b-a)E(U) = \frac{a+b}{2} \)$

Variance : $\( \text{Var}(X) = (b-a)^2\text{Var}(U) = \frac{(b-a)^2}{12} \)$