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

The exponential random variable has an interesting (and for a continuous r.v. unique!) property - called the memoryless property:
X has an exponential distribution iff X is a positive continuous r.v. and P(X>s+t | X>s) = P(X>t) for all s,t > 0.
Proof:
Assume X~E(λ). Then

on the other hand assume X is continuous with density f and P(X>s+t | X>s) = P(X>t) for all s,t > 0. In the proof above we saw that this implies P(X>s+t)=P(X>s)*P(X>t). Let h(x) = P(X>x) and let e>0. Note h(0) = P(X>0) = 1 because X is positive.

and so we see X~E(β)

The Gamma Distribution

Recall the gamma function:

The gamma function is famous for many things, among them the relationship G(α+1) = α G(α) which implies G(n)=(n-1)!. Also, we have G²(1/2) = π

Now X is said have a gamma distribution (X~G(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~G(1,β) then X~E(1/β).

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

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

The 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 open finite interval.

Special case: Beta(1,1) = U[0,1]

Let's go back to the gamma distribution for a moment. Say X and Y are independent G(α,β) and let Z=X+Y. Then

so we see that Z~G(2a,β). In other words, the sum of independent gamma r.v.'s is again Gamma.

Some special cases:
1) X,Y iid E(λ) then X+Y~G(2,λ) (and not exponential)
2) X,Y iid χ(n), then X+Y~χ(2n)

The 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

An Example of a Mixture of Discrete and Continous Distributions

Say we have a computer program that generates data as follows: first it generates a rv Z~Ber(p). It then generates a second rv X~N(μ·Z,1) that is X~N(0,1) if Z=0 and X~N(μ,1) if Z=1, for some μ.