Question

11

a) Find the conditional density of T; given that there are 10 arrivals in the time interval (0,1). b) Find the conditional density of Ts given that there are 10 arrivals in the time interval (0,1). c) Recognize the answers to a) and b) as named densities, and find the parameters. 11. Suppose X has uniform distribution on (-1,1) and, given X = 1, Y is uniformly distributed on (-V1-22. - 7?). Is (X,Y) then uniformly distributed over the unit disk {(1, y):r? + y2 < 1}? Explain carefully. 12. Suppose there are ten atoms, each of which decays by emission of an a-particle after an exponentially distributed lifetime with rate 1, independently of the others. Let Tibe the time of the first a-particle emission, T, the time of the second. Find: a) the distribution of T: b) the conditional distribution of T2 given T: c) the distribution of T2. 13. Let X and Y be independent random variables, X with uniform distribution on (0,3). SC AM

Answer 1

XnU(-1, 1) = fx (x) = pdf of x={ 1, if-1<x<1 Otherwise We also given that Ylx=x ~ U(-51-x², Ji-a2 =) po conditional pdf of Y given fix Cylx=x) = { - Jina? x=x x² sya <dina? otherwise Joint pdf of (+7) = fxxx (x, y) = x (x) + fylx_r (4)x=x) -(1-x²cycl-x? 4 - x² 1 - 1<x<i 0 otherwise x+y? <1,-KX<) 4 Jah otherwise a. (X,Y) not distributed Uniformly over ş (x,y); x74²<1} (proved),