# 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.

• 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(Xx) = (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 axb

## 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(Xx) = 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) = (xa)(1 / 𝑏−𝑎)

• Area to the Right of x**:** P(X > x) = (bx)(1 / 𝑏−𝑎)

• Area Between c and d**:** P(c < x < d) = (base)(height) = (dc)(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 XExp(m) where x ≥ 0 and m > 0

• exponential pdf: f(x) = me^(–mx) where x ≥ 0 and m > 0

• exponential cdf: P(Xx) = 1 – e^(–mx)

• exponential mean: µ = 1/𝑚

• exponential standard deviation: σ = µ

• exponential percentile k: k = 𝑙𝑛(1−𝐴𝑟𝑒𝑎𝑇𝑜𝑇ℎ𝑒𝐿𝑒𝑓𝑡𝑂𝑓𝑘) / (−𝑚)