customers arrive at a bank according to a poisson process with rate 10 per hour let nt be the nu

(a) The number of customers arriving in a given time period follows a Poisson distribution. The

Customers arrive at a bank according to a poisson process...

Customers arrive at a bank according to a Poisson process with rate λ = 10 per hour. Let N(t) be the number of customers up to time t. (a) Find the probability there are exactly two customers during the first half hour? (b) Given that 5 customers arrived during the first hour, find the probability that 3 arrived during the first half hour. (c) Suppose that customers come to the bank for three types of business, deposit, withdraw, open accounts, with respective probability 0.4, 0.4, 0.2. Find the probability that, among the first 5 customers, 2 of them come to open accounts.

Transcript

0:00 All right.

00:01 So here we have a poisson process about people arriving to bank.

00:08 We have lambda is 10 an hour.

00:11 And we're going to find the probability that there are exactly two customers during the first half hour.

00:15 So here's our formula here for the poisson distribution.

00:21 We have lambda.

00:22 I put l for lambda.

00:24 It's lambda to the x times e to the negative lambda.

00:27 All of our x factorial.

00:29 And i made a little function that takes in.

00:34 The lambda and the x value we want.

00:38 And so in this case, lambda is 10, but you're like, hey, guy, why, what's up with, i thought it was supposed to be 10, not 5.

00:46 Well, the reason for that is because this is for 10 per hour, but we're going to find two customers during the first half hour.

00:54 So if we have lambda is 10 per hour, 10 in one hour, oops, we want to get the equivalent in terms of a half hour.

01:06 So that would be equivalently five per half hour.

01:13 So 30 minutes.

01:20 You could do other kind of conversions, you know, 30 minutes into an hour and do stuff like that.

01:28 But, you know, because the poisson process takes about, thinks about the events per unit time.

01:36 So per hour.

01:37 So we need to convert this to half hour.

01:39 And so we want to know two and a half hour.

01:41 So we do is we plug 2 in for x 5 and for lambda and we get this value so the probability of 2 of x equals 2 is exact is a point or the x is exactly 2 is 0 .084 b given five customers arrived during the first half hour find the probability that 3 arrived during the first half hour all right so what do we here's this is what we have and the reason we have this is because we want um given this has occurred, given five arrived, that's what we're looking at.

02:25 So this is where five have arrived in an hour.

02:30 So this is...

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