Chapter 5: Continuous Random Variables

Probability density function

Probability density function

a statistical measure used to gauge the likely outcome of a discrete value

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.

Continuous probability distributions

PROBABILITY = AREA

Continuous probability density function

gives the relative likelihood of any outcome in a continuum occurring

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

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)

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)

Poisson probability

𝑃(𝑋=𝑘)=𝜆^𝑘 𝑒^−𝑘 / 𝑘! P(X=k) with mean λ

k!

k*(k-1)(k-2)(k-3)…32*1