6.4 Summary of the uniform and exponential probability distributions

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

Exponential

$X$ ~ $\mathrm{Exp}\left(m\right)$

$X$ = a real number, 0 or larger. $m$ = the parameter that controls the rate of decay or decline

The mean and standard deviation are the same.

$\mu =\sigma =\frac{1}{m}$ and $m=\frac{1}{\mu }=\frac{1}{\sigma }$

The probability density function: $f\left(X\right)=m\cdot e^{\mathrm{-m\cdot X}}$ , $X\ge 0$

Area to the Left of x: $P(X< x)=1-e^{\mathrm{-m\cdot x}}$

Area to the Right of x: $P(X> x)=e^{\mathrm{-m\cdot x}}$

Area Between c and d: $P(c< X< d)=P(X< d)-P(X< c)=(1-e^{\mathrm{-\; m\cdot d}})-(1-e^{\mathrm{-\; m\cdot c}})=e^{\mathrm{-\; m\cdot c}}-e^{\mathrm{-\; m\cdot d}}$

Percentile, k: $k=\frac{\text{LN(1-AreaToTheLeft)}}{\mathrm{-m}}$