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$X$ = 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: $P(c< X< d)=\left(\text{base}\right)\left(\text{height}\right)=(d-c)\left(\text{height}\right)$ .
Formula$X$ ~ $\mathrm{Exp}\left(m\right)$
$X$ = 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: $k=\frac{\text{LN(1-AreaToTheLeft)}}{\mathrm{-m}}$
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