Chapter 5: Continuous Random Variables

Introduction

Properties of Continuous Probability Distributions
  • Probability density function: a statistical measure used to gauge the likely outcome of a discrete value
      * f(x) ≥ 0
      * The total area under the curve f(x) is one.

     the formula for the probability density function of X

  • Cumulative distribution function: a function whose value is the probability that a corresponding continuous random variable has a value less than or equal to the argument of the function.
      * P(X ≤ x) = (x−a)/(b−a)

 

5.1 Continuous Probability Functions

  • Continuous probability distributions: PROBABILITY = AREA
  • Continuous probability density function: gives the relative likelihood of any outcome in a continuum occurring
      * f(x)=1 / b−a for a ≤ x ≤ b

5.2 The Uniform Distribution

  • The uniform distribution: is a continuous probability distribution and is concerned with events that are equally likely to occur.
      * Uniform Mean: 𝜇=(a+b) / 2
      * Uniform Standard deviation: 𝜎=√((b−a)^2 / 12)
      * Uniform pdf: f(x)=1 / b−a for a ≤ x ≤ b
      * Uniform cdf: P(X ≤ x) = x−a / b−a
  • 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
  • Probability density function: f(x)=(1 / b−a) for a ≤ X ≤ b
  • Area to the Left of x**:** P(X < x) = (x – a)(1 / b−a)
  • Area to the Right of x**:** P(X > x) = (b – x)(1 / b−a)
  • Area Between c and d**:** P(c < x < d) = (base)(height) = (d – c)(1 / b−a)

5.3 The Exponential Distribution

  • Memoryless property: the independence of events or, more specifically, the independence of event-to-event times or P (X > r + t | X > r) = P (X > t) for all r ≥ 0 and t ≥ 0
  • Exponential Distribution: X ~ Exp(m) where m = the decay parameter
      * decay parameter: m = 1 / μ and we write X ∼ Exp(m) where x ≥ 0 and m > 0
      * exponential pdf: f(x) = me^(–mx) where x ≥ 0 and m > 0
      * exponential cdf: P(X ≤ x) = 1 – e^(–mx)
      * exponential mean: µ = 1/m
      * exponential standard deviation: σ = µ
      * exponential percentile k: k = ln(1−AreaToTℎeLeftOfk) / (−m)
      * Additionally
        * P(X > x) = e^(–mx)
        * P(a < X < b) = e^(–ma) – e^(–mb)
  • Poisson probability:  P(X=k)=𝜆^k e^−k / k! P(X=k) with mean λ
  • k**!** = k*(k-1)*(k-2)*(k-3)*…3*2*1

Examples