A local fire station receives on average 85 emergency...

Name: a local fire station receives on average 85 emergency telephone calls per hour assume that these calls are poisson distributed calculate the probability that a the fire station will get nine 18273
Uploaded: 2022-05-29T21:35:59-08:00
Duration: 11 min 1 s
Channel: Derek Fairburn
Description: a local fire station receives on average 85 emergency telephone calls per hour assume that these calls are poisson distributed calculate the probability that a the fire station will get nine 18273

A local fire station receives on average 8.5 emergency telephone calls per hour. Assume that these calls are Poisson distributed. Calculate the probability that (a) the fire station will get nine calls for one hour. (2) (b) the fire station will get less than five calls for one hour. (2) (c) the fire station will get at least 4 calls for one hour. (2) (d) the fire station will get more than 6 calls for one hour. (2)

Transcript

00:01 A poisson distribution is used to measure the probability of a certain frequency of an event occurring in a given time period, given a mean number of, or given a mean frequency during the time period.

00:15 And so we're using this distribution for calls to a fire station that has a mean frequency of 8 .5 calls per hour.

00:27 So we calculate the poisson probabilities using the cumulative density function of the probability of x is lessen or equal to a certain value, is equal to the sum from zero to x of the following function.

00:51 So first we're going to do the probability of x equaling nine, or the fire station getting nine.

00:57 Calls in the hour.

00:59 So here we only have to evaluate the function at the actual value of x equals 9.

01:05 And so this is going to be equal to e to the negative 8 .5 times 8 .5 to the 9 divided by 9 factorial.

01:21 So i'm going to put this on my calculator really quick, and i will update with results.

01:30 And you get a probability of 0 .1 to 99.

01:57 Next we wanted to see inclusive, what the probability is that the fire station gets between 5 and 7 calls.

02:06 And so now we want to take the sum from 5 to 7 of this formula, e to the negative lambda, lambda to the i, divided by i factorial.

02:21 So this is equal to e to the negative 8 .5 times 8 .5 to the 5th divided by 5 factorial, plus e to the negative 8 .5, 8 .5 to the 6 divided by 6 factorial, plus e to the negative 8 .5, 8 .5, to the 7 divided by 7 factorial.

02:51 All right, so just give me a minute to put this in my calculator.

03:11 So we get individual probabilities of 0 .07523 plus 0 .1068 plus point 1 -2942.

03:58 And so if we add these all together, we get a probability that the number of calls is between 5 and 7 .7.

04:11 Inclusive of 0 .3112.

04:18 The next is asking, what is the probability that the station gets at least four calls? so that includes four.

04:27 This is equal to 1 minus the probability that x is less than 4.

04:36 Because theoretically the station could get 100 calls, that's very unlikely.

04:41 This keeps going infinitely.

04:45 It's more practical to rewrite it in this form so that we can explicitly calculate the probability that x equals 0 plus the probability that x equals 1 plus the probability that x equals 2 plus the probability that x equals 3.

05:07 And so for the probability that x equals 0, it's just going to be e to the negative 8...

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