6.4 Summary of the uniform distribution

Uniform

$X$ = a real number between $a$ and $b$ (in some instances, $X$ can take on the values $a$ and $b$ ). $a$ = smallest $X$ ; $b$ = largest $X$

$X$ ~ $U\left(\mathrm{a,}b\right)$

The mean is $\mu =\frac{a+b}{2}$

The standard deviation is $\sigma =\sqrt{\frac{(b-a{)}^2}{12}}$

Probability density function: $f(X)=\frac{1}{b-a}$ for $a\le X\le b$

Area to the Left of x: $P(X< x)=\text{(base)}\text{(height)}$

Area to the Right of x: $P(X> x)=\text{(base)}\text{(height)}$

Area Between c and d: $P(c< X< d)=\left(\text{base}\right)\left(\text{height}\right)=(d-c)\left(\text{height}\right)$ .