Recent content by Mehmood_Yasir

This makes me think again for my own problem .amazing. as I said explicitly in above post#3 with an example that assuming 4 pedestrian arrive within the time window ##T## which is started by the first one. So, I am looking to find the waiting time distributions of pedestrians who are now going...

Homework Statement Pedestrians approach to a signal for road crossing in a Poisson manner with arrival rate ##\lambda## per sec. The first pedestrian arriving the signal pushes the button to start time ##T##, and thus we assume his arrival time is ##t=0##, and he always see ##T## wait time. A...

Actually I have not that much idea how to assess the accuracy of approximation, need to explore further, but I just developed a program that runs this process for 1 million time, interesting the empirical cdf of ##W_k## matches cdf of my approximation. would it be enough to verify the...

Hmm, I have been thinking over the night about it, what is your opinion if I say that, consider only a time interval ##0## to ##T##, first pedestrian pushes button to start the process, now forget about this pedestrian. The remaining pedestrians arrive following poisson process with rate...

when is it going to end to get the pdf or cdf of ##W_k##?

It is ##C## and since it is not 1. Do I need to multiply ##\frac{1}{C}## with ##\frac d {dw} P(W_k \leq w | W_k \leq T)= \frac{ \frac{ \lambda^k {(T-w)}^{k-1} e^{-\lambda (T-w)} }{(k-1)!} }{(1-e^{-\lambda T)}} ##

there is a minus sign as well, - ##\sum_{m=0}^{k-1} \frac{{(\lambda (T))}^m e^{-\lambda (T)}}{m!}##

by inserting upper limit ##T## in the sum - lower limit ##0## in the sum gives ##\sum_{m=0}^{k-1} \frac{{(\lambda (T))}^m e^{-\lambda (T)}}{m!}##

this is a long go..I tried as it looks never ending integral,

right, yes It is in fact ##f_{t_k|t_k \leq T} (t)## what i am plotting

Limits should be from ##0## to ##T## for integr. (What limits do you have to evaluate ##\sum_{m=0}^{k-1} \frac{{(\lambda (T-w))}^m e^{-\lambda (T-w)}}{m!}## at?)

I am a bit confused as I am plotting the pdf of ##W_k## and ##t_k## in MATLAB for the above process. For the time vector ##t## on X-axis, I plot the pdf of ##t_k## using its density function ##f_{t_k} (t)= \frac{ {(\lambda t)}^k e^{-\lambda t}} {k!}## for time vector ##0\leq t \leq T##. Thats...

##P(W_k \leq w)=1-P(t_k < T-w)## ##P(W_k \leq w)=\sum_{m=0}^{k-1} \frac{{(\lambda (T-w))}^m e^{-\lambda (T-w)}}{m!}##

for plotting pdf of ##t_k## and pdf of ##W_k## against a time vector e.g., ##t=0:0.01:T##(minutes) on x axis, I need to replace ##w## in pdf of ##W_k## with ##t## such that ##f_{W_k} (t)= \frac{ \frac{ \lambda^k {(T-t)}^{k-1} e^{-\lambda (T-t)} }{(k-1)!} }{(1-e^{-\lambda T)}} ##...

That's correct. O sorry. Actual result is ##= \frac{ \lambda^k {(T-w)}^{k-1} e^{-\lambda (T-w)} }{(k-1)!}## Yes, ##= \frac{ \frac{ \lambda^k {(T-w)}^{k-1} e^{-\lambda (T-w)} }{(k-1)!} }{1-P(W_k>T)} ## ##= \frac{ \frac{ \lambda^k {(T-w)}^{k-1} e^{-\lambda (T-w)} }{(k-1)!}...