Chapter 8 Special Continuous Distributions

8.1 Continuous Uniform Random Variables

XUnif(a,b)

f(x)={1baif axb0otherwise

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)=(ba)212σx=ba12

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.

XExponential(λ)orXExponential(θ=1λ)

fX(x)=F(x)={λeλxif x00x<0

Center and Dispersion

E(X)=σx=1λVar(X)=1λ2

Additional Notes

  • Memoryless property
  • Relationship between Exponential and Geometric

fX(x)=F(x)={λeλxif x00x<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.

ZN(0,1)

f(z)=12πexp[z22]

F(z)=12πzexp[t22]dt

Center and Dispersion

E(X)=0Var(X)=1

Additional Notes

  • We standardize

8.3.2 Normal Random Variable

XN(μ,σ2)

f(x)=1σ2πexp[(xμ)22σ2]<x<

Center and Dispersion

E(X)=μVar(X)=σ2

Additional Notes