Continuous Distributions

Uniform Distribution

X is said to have a uniform distribution on the interval [A,B] if

Exponential Distribution

X is said to have an exponential distribution rate λ if

Gamma Distribution

Recall the gamma function:

The gamma function is famous for many things, among them the relationship Γ(α+1) = α Γ(α) which implies Γ(n)=(n-1)!. Also, we have Γ²(1/2) = π

Now X is said have a gamma distribution (X~Γ(α,β)) with parameters (α,β) if

By definition we have X>0, and so the Gamma is the basic example of a r.v. on (0,∞), or a little more general (using a change of variables) on any open half interval

Note if X~Γ(1,β) then X~E(1/β).

Another important special case is if X~Γ(n/2,2), then X is called a Chi-square r.v. with n degrees of freedom, denoted by X~ χ²(n)

There is an important connection between the Gamma and the Poisson distributions:
If X~Γ(n,β) and Y~P(x/β) then
P(X≤x) = P(Y≥n)

Beta Distribution

X is said to have a beta distribution with parameters α and β (X~Beta(α,β)) if

By definition we have 0<X<1, and so the Beta is the basic example of a r.v. on [0,1], or a little more general (using a change of variables) on any finite interval.

Some special cases:

1) Beta(1,1) = U[0,1]
2) X,Y iid E(λ) then X+Y~Γ(2,λ) (and not exponential)
3) X~χ²(n), Y~χ²(m), XY then X+Y~χ²(n+m)

Normal (Gaussian) Distribution

X is said to have a normal distribution with mean μ and variance σ² (X~N(μ,σ)) if it has density

Of course we have EX=μ and V(X)=σ²

We also have the following interesting features:

1) If X ~ N(μ_X,σ_X²) and Y~N(μ_Y,σ_Y²) then

Z = X + Y ~ N(μ_X+μ_Y,σ_X²+σ_Y²+2σ_Xσ_Yρ)

where ρ = cor(X,Y)

2) if cov(X,Y) = 0 then X and Y are independent

3) P(X>μ) = P(X<μ) =1/2

4) P(X>μ+x) = P(X<μ-x)

5) say X~N(μ,σ) then