Sections
Formula Review

# Formula Review

### 5.1Continuous 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(Xx)

### 5.2The 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 $μ=a+b2.μ=a+b2.$

The standard deviation is

Probability density function: $f(x)=1b−af(x)=1b−a$ for $a≤X≤ba≤X≤b$

Area to the left of x: P(X < x) = (xa)$(1b−a)(1b−a)$

Area to the right of x: P(X > x) = (bx)$(1b−a)(1b−a)$

Area between c and d: P(c < x < d) = (base)(height) = (dc)$(1b−a)(1b−a)$

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

• pdf: $f(x)=1b−af(x)=1b−a$ for a ≤ x ≤ b
• cdf: P(Xx) = $x−ab−ax−ab−a$
• mean µ = $a+b2a+b2$
• standard deviation σ $=(b−a)212=(b−a)212$
• P(c < X < d) = (dc)$(1b–a)(1b–a)$

### 5.3The 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(Xx) = 1 – e(–mx)
• mean µ = $1m1m$
• standard deviation σ = µ
• percentile k: k = $ln(1−AreaToTheLeftOfk)(−m)ln(1−AreaToTheLeftOfk)(−m)$