Question

In: Statistics and Probability

Assume that the Poisson process X = {X(t) : t ≥ 0} describes students’ arrivals at...

Assume that the Poisson process X = {X(t) : t ≥ 0} describes students’ arrivals at the library with intensity λ = 4 per hour. Given that the tenth student arrived exactly at the end of fourth hour, or W10 = 4, find:

1. E [W1|W10 = 4]

2. E [W9 − W1|W10 = 4].

Hint: Suppose that X {X(t) : t ≥ 0} is a Poisson process with rate λ > 0 and its arrival times are defined for any natural k as Wk = min[t ≥ 0 : X(t) = k] (1) Then for any natural m, the inter-arrival times, {T1 = W1, T2 = W2 − W1, . . . , Tm = Wm − Wm−1} are independent variables with the common exponential distribution, fT(t) = λ · e −λ·t for t > 0.

Solutions

Expert Solution

orchestra answered 1 year ago
Next > < Previous

Related Solutions

Students enter the bathroom according to a Poisson process at a rate of 7.5 arrivals per...
Students enter the bathroom according to a Poisson process at a rate of 7.5 arrivals per minute. What is the probability that exactly 46 students enter between 3:00 and 3:05? Given that 6 students enter the bathroom between 4:00 and 4:01, what is the probability that exactly 36 students enter between 4:00 and 4:07? Each student entering the bathroom has a .15 probability of wearing a hoodie, independent of other students. What is the probability that exactly 10 students wearing...
Consider a homogeneous Poisson process {N(t), t ≥ 0} with rate α. Now color each point...
Consider a homogeneous Poisson process {N(t), t ≥ 0} with rate α. Now color each point blue with probability p and red with probability q = 1 − p. Colors of distinct points are independent. Let B2 be the location of the 2nd blue point. Find E(B2).
show that if {X (t), t≥ 0} is a stochastic process with independent processes, then the...
show that if {X (t), t≥ 0} is a stochastic process with independent processes, then the process defined by Y (t) = X (t) - X (0), for t≥ 0, has independent increments and Y(0)=0
If X(t), t ≥ 0 is a Brownian motion process with drift parameter μ and variance...
If X(t), t ≥ 0 is a Brownian motion process with drift parameter μ and variance parameter σ2 for which X(0)=0, show that -X(t), t ≥ 0 is a Brownian motion process with drift parameter -μ and variance parameter σ2.
(9) Assume on average 10 passengers arrive per minute. Assuming poisson arrivals and departures, estimate the...
(9) Assume on average 10 passengers arrive per minute. Assuming poisson arrivals and departures, estimate the gain (if any) in ‘average time spent in system per passenger’ if TSA decides to replace 4 type-A security scanners with 3 type-B security scanners. The service rate per scanner for type-A scanners is 3 passengers per minute and type-B scanners is 5 passengers per minute?
y''(t)+(x+y)^2*y(t)=sin(x*t+y*t)-sin(x*t-y*t), y(0)=0, y'(0)=0, x and y are real numbers
y''(t)+(x+y)^2*y(t)=sin(x*t+y*t)-sin(x*t-y*t), y(0)=0, y'(0)=0, x and y are real numbers
Please solve the following: ut=uxx, 0<x<1, t>0 u(0,t)=0, u(1,t)=A, t>0 u(x,0)=cosx, 0<x<1
Please solve the following: ut=uxx, 0<x<1, t>0 u(0,t)=0, u(1,t)=A, t>0 u(x,0)=cosx, 0<x<1
x"(t)- 4x'(t)+4x(t)=4e^2t ; x(0)= -1, x'(0)= -4
x"(t)- 4x'(t)+4x(t)=4e^2t ; x(0)= -1, x'(0)= -4
Solve the nonhomogeneous heat equation: ut-kuxx=sinx, 0<x<pi, t>0 u(0,t)=u(pi,t)=0, t>0 u(x,0)=0, 0<x<pi
Solve the nonhomogeneous heat equation: ut-kuxx=sinx, 0<x<pi, t>0 u(0,t)=u(pi,t)=0, t>0 u(x,0)=0, 0<x<pi
utility function u(x,y;t )= (x-t)ay1-a x>=t, t>0, 0<a<1 u(x,y;t )=0 when x<t does income consumption curve...
utility function u(x,y;t )= (x-t)ay1-a x>=t, t>0, 0<a<1 u(x,y;t )=0 when x<t does income consumption curve is y=[(1-a)(x-t)px]/apy ?(my result, i used lagrange, not sure about it) how to draw the income consumption curve?
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT