Chapter 8 Special Continuous Distributions
8.1 Continuous Uniform Random Variables
X∼Unif(a,b)
f(x)={1b−aif a≤x≤b0otherwise
X is uniformly distributed over the interval [a,b], so at all points we have the same height for the PDF. Any two intervals of the same length can occur with equal probability.
Center and Dispersion
E(X)=a+b2Var(X)=(b−a)212σx=b−a√12
Additional Notes
There is also a discrete uniform random variable which is similarly defined.
A piece-wise constant pdf has a total area of 1, but can have discontinuities.
8.2 Exponential Random Variables
Models the inter-arrival time as a random variable, or amount of time until an event occurs.
X∼Exponential(λ)orX∼Exponential(θ=1λ)
fX(x)=F′(x)={λe−λxif x≥00x<0
Center and Dispersion
E(X)=σx=1λVar(X)=1λ2
Additional Notes
- Memoryless property
- Relationship between Exponential and Geometric
fX(x)=F′(x)={λe−λxif x≥00x<0
8.3 Gaussian Random Variables
8.3.1 Standard Normal Random Variable
A special case of the normal distribution that is used to calculated probabilities for any normal distribution, without having to calculate using it’s PDF.
Z∼N(0,1)
f(z)=1√2πexp[−z22]
F(z)=1√2π∫z−∞exp[−t22]dt
Center and Dispersion
E(X)=0Var(X)=1
Additional Notes
- We standardize