Stat 5101 Lecture Notes - School of Statistics

4.4. THE POISSON PROCESS 119Hence the Jacobian isJ(u, v) =∣∂x∂u∂y∂u∂x∂v∂y∂v∣ ∣∣∣∣∣ ∣ = v u1−v −u∣ = −uThe joint density of X and Y is f X (x)f Y (y) by independence. By the changeof variable formula, the joint density of U and V isf U,V (u, v) =f X,Y [uv, u(1 − v)]|J(u, v)|= f X (uv)f Y [u(1 − v)]u= λsλtΓ(s) (uv)s−1 e −λuvΓ(t) [u(1 − v)]t−1 e −λu(1−v) u= λs+tΓ(s)Γ(t) us+t−1 e −λu v s−1 (1 − v) t−1Since the joint density factors into a function of u times a function of v, thevariables U and V are independent. Since these functions are proportional tothe gamma and beta densities asserted by the theorem, U and V must actuallyhave these distributions.Corollary 4.3.B(s, t) = Γ(s)Γ(t)Γ(s + t)Proof. The constant in the joint density found in the proof of the theorem mustbe the product of the constants for the beta and gamma densities. Henceλ s+tΓ(s)Γ(t) =Solving for B(s, t) gives the corollary.λs+t 1Γ(s + t) B(s, t)For moments of the beta distribution, see Lindgren pp. 176–177.4.4 The Poisson Process4.4.1 Spatial Point ProcessesA spatial point process is a random pattern of points in a region of space.The space can be any dimension.A point process is simple if it never has points on top of each other so thateach point of the process is at a different location in space. A point process isboundedly finite if with probability one it has only a finite number of points inany bounded set.Let N A denote the number of points in a region A. Since the point patternis random, N A is a random variable. Since it counts points, N A is a discrete