Formula Review

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(Xx)

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.μ=a+b2.

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

Probability density function: f(x)=1baf(x)=1ba for aXbaXb

Area to the left of x: P(X < x) = (xa)(1ba)(1ba)

Area to the right of x: P(X > x) = (bx)(1ba)(1ba)

Area between c and d: P(c < x < d) = (base)(height) = (dc)(1ba)(1ba)

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

  • pdf: f(x)=1baf(x)=1ba for a ≤ x ≤ b
  • cdf: P(Xx) = xabaxaba
  • mean µ = a+b2a+b2
  • standard deviation σ =(ba)212=(ba)212
  • P(c < X < d) = (dc)(1ba)(1ba)

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(Xx) = 1 – e(–mx)
  • mean µ = 1m1m
  • standard deviation σ = µ
  • percentile k: k = ln(1AreaToTheLeftOfk)(m)ln(1AreaToTheLeftOfk)(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)=λkekk! P(X=k)=λkekk! with mean λ
  • k! = k*(k−1)*(k−2)*(k−3)*…3*2*1