2. (10 points) Cars arrive at a toll booth at a mean rate of...

Name: 2. (10 points) Cars arrive at a toll booth at a mean rate of five cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.3 minutes before collecting the eighth toll.
Uploaded: 2022-02-20T11:24:15-08:00
Duration: 5 min 18 s
Channel: Setu R
Description: 2. (10 points) Cars arrive at a toll booth at a mean rate of five cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.3 minutes before collecting the eighth toll.

2. (10 points) Cars arrive at a toll booth at a mean rate of five cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.3 minutes before collecting the eighth toll.

Transcript

00:01 Hello student in this question we have x as a poison process okay with the parameter as lambda which is equals to mu means we can say five cars over 10 minutes okay we can simplify it one over two cars per minute okay now we have w as the waiting time here okay and we can say until lambda occurrence okay and it follows we can say it follows gamma distribution with the parameters we know the parameters of gamma functions that lambda and theta which is equal to one over lambda alpha and 1 is theta which is equal to 1 over lambda so now we have alpha is 8 and the value of theta is 1 over lambda means 1 over 1 by 2 so it is 2 now what we'll calculate is a probability of waiting time greater than 26 .30 okay which is 1 minus probability of of waiting time less than or equals to 26 .30.

01:41 The formula for calculating the probability of waiting time less than or equals to 26 .30 is 1 minus submission k 0 to alpha minus 1 lambda omega power k e rase 4 minus lambda omega over k factor we'll use this formula here.

02:09 It becomes 1 minus submission k0 up to 8 minus 1 .1 minus 1 .1 .1 .2 into 26 .30.

02:23 Okay, power both k and into e ristfer minus 1 by 2 into 26 .30.

02:33 And whole denominator by k factorial okay now here break it close you can say that one in one you can say zero and this minus and minus becomes plus we can write here zero plus summation of k zero up to seven and here when we multiply it becomes 13 .15 power k e r s to par minus 13 .15 over k factorial okay now we will open this we will remain with values 13 .15 power 0 e raced by minus 13 .15 over 0 factorial okay then adding 13 .15 power 1 e raced per minus 13 .15 .15 denominator 1 factorial then adding 13 .15 power 2 a r .5 .15 over 2 factorial plus 13 .15 power 3 e...

Question

2. (10 points) Cars arrive at a toll booth at a mean rate of five cars every 10 minutes according to a Poisson process. Find the probability that the toll collector will have to wait longer than 26.3 minutes before collecting the eighth toll.

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