Chapter 5: Continuous Random Variables

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)*…3*2\*1