Mathematics Homework Help

MAT 135 A — Luis RademacherMidterm 2 — December 11th — 50 minutesOpen book and open notes. Every student must write his or her own solutions; looking forsolutionsfrom external sources (the web, material from previous years, etc.) is prohibited.Justify or prove all your answers.1. Let (X,Y), (U,V) be a pair of independent random points in square [−1,1]2. LetD= (X−U)2+ (Y−V)2be the squared distance between them. Determine(a) E(D) and(b) var(D).1

2. LetNbe a Poisson random variable with parameterλ. LetX1,…,XNbeNiid.random variables following the standard Gaussian distribution. Determine E(X21+···+X2N).2

3. Letabe an unknown length that you are trying to measure. You take measure-mentsntimes with an instrument. Measurementihas valuea+Ei, that is, it hasmeasurement errorEi,i= 1,…,n.E1,…,Enare iid and each follows a standardGaussian distribution. You estimate the length by the average of the measurements,X= (1/n)∑ni=1a+Ei. Write Chebyshev’s inequality specialized toXto bound theprobability thatXis not in (a−,a+).3

4. Let (X,Y) be a random point in the unit disk{(x,y)∈R2:x2+y2≤1}.(a) Determine the marginal density ofX.(b) DetermineP(X≤Y).(c) Determine the conditional density ofXgivenY=y.(d) AreX,Ymutually independent