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 μ=a+b2

$�=\frac{�+�}{2}$ 

The standard deviation is σ=(b – a)212−−−−−√

$�=\sqrt{\frac{{(�\text{ – }�)}^2}{12}}$ 

Probability density function: f(x)=1b−a

$�(�)=\frac{1}{�-�}$ for a≤X≤b

$�\le �\le �$ 

Area to the Left of x: P(X < x) = (x – a)(1b−a)

$\left(\frac{1}{�-�}\right)$ 

Area to the Right of x: P(X > x) = (b – x)(1b−a)

$\left(\frac{1}{�-�}\right)$ 

Area Between c and d: P(c < x < d) = (base)(height) = (d – c)(1b−a)

$\left(\frac{1}{�-�}\right)$ 

• pdf: f(x)=1b−a
  $�\left(�\right)=\frac{1}{�-�}$ 

  for a ≤ x ≤ b

• cdf: P(X ≤ x) = x−ab−a
  $\frac{�-�}{�-�}$ 
• mean µ = a+b2
  $\frac{�+�}{2}$ 
• standard deviation σ =(b−a)212−−−−−√
  $=\sqrt{\frac{{(�-�)}^2}{12}}$ 
• P(c < X < d) = (d – c)(1b–a)
  $(\frac{1}{�–�})$ 

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 = 1m
  $\frac{1}{�}$ 
• standard deviation σ = µ
• Additionally
  • P(X > x) = e^(–mx)
  • P(a < X < b) = e^(–ma) – e^(–mb)
• Poisson probability: P(X=x)=μxe−μx!
  $�(�=�)=\frac{{�}^{�}{�}^{-�}}{�!}$ 

  with mean and variance of μ