1. Customers arrive at a service facility according to a...

Name: 1. Customers arrive at a service facility according to a Poisson process of rate ? = 5 per hour. Let Nt be the number of customers that have arrived up to time t (hours). Determine the following probabilities. a. P(N1 = 5). b. P(N1 = 5 and N3 = 12). c. P(N3 = 12 | N1 = 5). d. P(N1 = 5 | N3 = 12).
Uploaded: 2023-07-13T14:08:35-08:00
Duration: 6 min
Channel: Madhur L
Description: 1. Customers arrive at a service facility according to a Poisson process of rate ? = 5 per hour. Let Nt be the number of customers that have arrived up to time t (hours). Determine the following probabilities. a. P(N1 = 5). b. P(N1 = 5 and N3 = 12). c. P(N3 = 12 | N1 = 5). d. P(N1 = 5 | N3 = 12).

1. Customers arrive at a service facility according to a Poisson process of rate ? = 5 per hour. Let Nt be the number of customers that have arrived up to time t (hours). Determine the following probabilities. a. P(N1 = 5). b. P(N1 = 5 and N3 = 12). c. P(N3 = 12 | N1 = 5). d. P(N1 = 5 | N3 = 12).

Transcript

00:01 Hello students, it is given that the customers arrive at a service facility according to a poisson process of rate lambda is equal to 5 per hour and also given that nt be the number of customers that have arrived up to time t, then we have to determine the following probabilities.

00:20 Now, let xi be the number of customers, number of customers arrived per ith hour, which implies n1 is equal to x1, n2 is equal to x1 plus x2, then n3 is equal to x1 plus x2 plus x3 and so on.

00:58 Now, when xi's are independently distributed, independently distributed, then we can calculate the probability of n1 is equal to 5, which is same as probability of x1 is equal to 5 and now by using the poisson formula, we will get this is e raised to minus 5, since we have given the parameter lambda is equal to 5 into 5 raised to 5 and divide by 5 factorial that is equal to 0 .1755.

01:42 This is the required probability.

01:45 Then in the second question, we have to calculate probability of n1 is equal to 5 and n3 is equal to 12.

01:56 So now this is equal to probability of x1 is equal to 5 and n3 that means x1 plus x2 plus x3 which is equal to 12.

02:10 Now, since we have given that x1 is equal to 5, that means we can write probability of x1 is equal to 5 and x2 plus x3 is equal to 12 minus 5, which is equal to 7 and since we have given that x1, x2, x3 are independent, we can write this as probability of x1 is equal to 5 into probability of x2 plus x3 is equal to 7 and this is equal to e raised to minus 5 into 5 raised to 5 divide by 5 factorial into e raised to minus 10 into 10 raised to 7 divide by 7 factorial and that is equal to 0 .0158.

02:58 This is the required probability.

03:01 Then in the next question, we have to calculate probability of n3 is equal to 12 given n1 is equal to 5.

03:13 Now by using the formula for the conditional probability, this we can write it as probability of n3 is equal to 12, n1 is equal to 5 divide by probability of n1 is equal to 5.

03:29 So now this is equal to probability of x1 plus x2 plus x3 is equal to 12, x1 is equal to 5 divide by probability of x1 is equal to 5 and now this is equal to probability of x2 plus x3 is equal to 7 into probability of x1 is equal to 5 and that is divide by probability of x1 is equal to 5...

Question

1. Customers arrive at a service facility according to a Poisson process of rate ? = 5 per hour. Let Nt be the number of customers that have arrived up to time t (hours). Determine the following probabilities. a. P(N1 = 5). b. P(N1 = 5 and N3 = 12). c. P(N3 = 12 | N1 = 5). d. P(N1 = 5 | N3 = 12).

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