### 5.1 Properties of Continuous Probability Density Functions

Probability density function (pdf) *f*(*x*):

*f*(*x*) ≥ 0- The total area under the curve
*f*(*x*) is one.

Cumulative distribution function (cdf): *P*(*X* ≤ *x*)

### 5.2 The Uniform Distribution

*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* (a, b)

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

The standard deviation is $\sigma =\sqrt{\frac{{(b\text{\u2013}a)}^{2}}{12}}$

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

**Area to the Left of x:**

*P*(

*X*<

*x*) = (

*x*–

*a*)$\left(\frac{1}{b-a}\right)$

**Area to the Right of x:**

*P*(

*X*>

*x*) = (

*b*–

*x*)$\left(\frac{1}{b-a}\right)$

**Area Between c and d:**

*P*(

*c*<

*x*<

*d*) = (base)(height) = (

*d*–

*c*)$\left(\frac{1}{b-a}\right)$

- pdf: $f\left(x\right)=\frac{1}{b-a}$
for
*a ≤ x ≤ b* - cdf:
*P*(*X*≤*x*) = $\frac{x-a}{b-a}$ - mean
*µ*= $\frac{a+b}{2}$ - standard deviation
*σ*$=\sqrt{\frac{{(b-a)}^{2}}{12}}$ *P*(*c*<*X*<*d*) = (*d*–*c*)$(\frac{1}{b\u2013a})$

### 5.3 The Exponential Distribution

- pdf:
*f*(*x*) =*me*^{(–mx)}where*x*≥ 0 and*m*> 0 - cdf:
*P*(*X*≤*x*) = 1 –*e*^{(–mx)} - mean
*µ*= $\frac{1}{m}$ - standard deviation
*σ*=*µ* - Additionally
*P*(*X*>*x*) =*e*^{(–mx)}*P*(*a*<*X*<*b*) =*e*^{(–ma)}–*e*^{(–mb)}

- Poisson probability: $P(X=x)=\frac{{\mu}^{x}{e}^{-\mu}}{x!}$ with mean and variance of
*μ*