5.1 Continuous Probability 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 $\mu =\frac{a+b}{2}\text{.}$

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

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

Uniform: X ~ U(a, b) where a < x < b

• 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)$(\frac{1}{b–a})$

5.3 The Exponential Distribution (Optional)

Exponential: X ~ Exp(m) where m = the decay parameter

• 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
• percentile k: k = \frac{ln(1-AreaToTheLeftOfk)}{(-m)}
• Additionally:
  • P(X > x) = e^(–mx)
  • P(a < X < b) = e^(–ma) – e^(–mb)
• Memoryless property: P(X > x + k|X > x) = P (X > k)
• Poisson probability: $P(X=k)=\frac{{\lambda }^k{e}^{-k}}{k!}$ with mean λ
• k! = k*(k−1)*(k−2)*(k−3)*…3*2*1