Problem 0.1 Let X be the number of people who enter a bank by time t>0....
Problem o.1 Let X, be the number of people who enter a bank by time t > 0. Suppose k! for k- 0,1,2,..., and s (t - s)k-e-t for t>s> 0, and k2r 0,1,2,.... (a) Find Pr[X2 k| X 1 for k 0,1,2,.... (b) Find E2 X1 1 Useful information: Don't eat yellow snow, andeot/k! Problem o.2 Recall the Geometric(p) distribution where X- number of flips of a coin until you get a head (H) with Pr(H) - p. The...
Problem 0.2 Recall the Geometric(p) distribution where X-number of flips of a coin until you get a head (H) with Pr(H) -p. The distribution is Pr(X- (1-p)1p for 1,2,. , with mean E(X)x(1 - p)*-p- 1/p, which can be obtained by brute force. An easier way to find the mean is to condition on the first toss, say Y- 0or 1 if the first toss is T or H. Show the mean is 1/p using E(X) EE(X Y)
Recall the Geometric(p) distribution where X = number of flips of a coin until you get a head (H) with Pr(H) = p. The distribution is Pr(X = x) = (1 − p) (x−1) p for x = 1, 2, . . . , with mean E(X) = ∑ x=1∞ (x(1 − p) (x−1) p) = 1/p, which can be obtained by brute force. An easier way to find the mean is to condition on the first toss, say Y...
Problem 0.1 Let Xt be the number of people who enter a bank by time t > 0. Suppose Pr[Xt = k] ... Problem 0.1 Let Xt be the number of people who enter a bank by time t > 0. Suppose Pr[Xt = k] = (tk e−t )/k! , for k = [0, 1, 2, . . . ,] and Pr[Xt = k, Xs = r] = sr *(t − s)k−r *e−t /(r!(k − r)!) , for t >...
Problem 0.1 Let Xt be the number of people who enter a bank by time> 0. Suppose t*e-! for k = 0,1,2 ,and for t > s > 0, and k > r = 0,1,2, (b) Find ElXyIX,-1]. Useful information: Don't eat yellow snow, and e- .*/k!
Problem 0.1 Let Xt be the number of people who enter a bank by time t > 0. Suppose Pr[Xt = k] = (t k e −t )/k! , for k = [0, 1, 2, . . . ,] and Pr[Xt = k, Xs = r] = sr *(t − s)k−r *e−t /(r!(k − r)!) , for t > s > 0, and k ≥ r = 0, 1, 2, . . . . (a) Find Pr[X2 = k |...
1. Let {Xt;t >0} be a pure birth process with rate 1x > 0, for x € S = {0,1,2,...}. (a) Write the backward equations (KBE) and use it to solve for Prz(t). (b) Use the result to part (a) to show that the waiting time in state x, say Wx, is exponentially distributed (c) Suppose 1x = 1 is constant for all x E S. Prove by induction that Px-kx(t) = (at) ke Af/k! for k = 0,..., and...
Find the solution to the heat equation on the infinite domain ∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,0,|x|<1|x|>1.∂u∂t=k∂2u∂x2,−∞<x<∞,t>0,u(x,0)={x,|x|<10,|x|>1. in terms of the error function. Q1 (10 points) Find the solution to the heat equation on the infinite domain azu ди at k -00<x<0, t>0, ar2 u(x,0) (X, 1x < 1 10, [] > 1. in terms of the error function. + Drag and drop your files or click to browse...
Let X and Y have joint pdf f(x,y)=k(x+y), for 0<=x<=1 and 0<=y<=1. a) Find k. d) Find Pr[Y<X2] and Pr[X+Y>0.5]
4. Let 8 >0. Let X, X2,..., X, be a random sample from the distribution with probability density function S(*;ð) - ma t?e-vor x>0, zero otherwise. Recall: W=vX has Gamma( a -6, 0-ta) distribution. Y=ZVX; = Z W; has a Gamma ( a =6n, = ta) distribution. i=1 E(Xk) - I( 2k+6) 120 ok k>-3. 42 S. A method of moments estimator of 8 is 42.n 8 = h) Suggest a confidence interval for 8 with (1 - 0) 100%...