Question

differential equations proof please

Consider throwing a dart at the origin of the Cartesian plane. You are aiming at the origin, but random errors in your throw will produce varying results. We assume that: 1.) The errors do not depend on the orientation of the coordinate system. 2.) Errors in perpendicular directions are independent. This means that being too high doesn't alter the probability of being off to the right. 3.) Large errors are less likely than small errors. Let f(z, y) be a differentiable probability density function describing the above scenario. By defi- nition, this means that of(z, yardy 1. The three basic assumptions above translate to the following three axioms about f(x, g). A1: f(z,y)-f(xi,vi) whenever vi A2: f(x,y) f(x, 0)f(0, y) whenever0 and y o. A3: f(z, v) < f(zi, yi) whenever r' + y2 4 z + y? Our objective is to classify all functions f(x, y) that satisfy A1, A2 and A3. The following ex- ercises will guide you through how we go about finding f(z,y). Let С,-( (z, y) : z? + y2-p2} . For for each (zw) E C, define h(r):-f(z, y). Exercise 1:(5 points) Explain why h(r) is a well defined function Exercise 2:(7.5 points) Prove that f(z,0-f(0,z) for all z. (Hint: use A1) Let Exercise 3:(7.5 points) Prove that f(x,y)-f(x)f(a) and h(r) f(x)f(w) (Hint:Use A2 and Exercise 2). Exercise 4:(7.5 points) Prove that -Ди whenever (z,y) E C, and z 0 and y 0. Hint: Using the chain rule, take the derivative of h(r)-f(a)f(v) with respect to z and then go back and take the derivative of h(r) with respect to y). Exercise 5:(7.5 points) Prove that ,ー=constant for all 。 Exercise 6:(7.5 points) Use the DE in Exercise 5 to solve for the general solution of flx). Exercise 7:(7.5 points) Find conditions on the general solution of f(z) so that (Hint: Change to polar coordinates and integrate, do not forget the Jacobian.)

Answer 1

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