Traffic flow is traditionally modeled as a Poisson distribution. A traffic engineer monitors the

step 1 So traffic flow is traditionally modeled by Poisson distribution. So we are going to be using Po

Traffic flow is traditionally modeled as a Poisson...

Key Concepts

Variance-Mean Relationship in Poisson Distribution

A key property of the Poisson distribution is that its variance is equal to its mean. This characteristic can serve as a check for the appropriateness of applying the Poisson model. Deviations from this relationship, such as significantly higher variance than the mean, can indicate that the data may not be well-modeled by a Poisson process and that alternative models should be considered.

Cumulative Distribution Function (CDF)

The cumulative distribution function for a Poisson distribution provides the probability that the random variable takes on a value less than or equal to a given number. This is particularly useful for determining the minimum count required to achieve a certain level of confidence or cumulative probability, often needed when setting thresholds or bounds in practical applications.

Poisson Process

A Poisson process is a statistical model used to describe events happening randomly over time or space, where the events occur independently of one another. In a Poisson process, the number of events in any interval of time follows a Poisson distribution with the mean proportional to the length of the interval. This process underpins the use of the Poisson distribution in various fields, including traffic flow modeling.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that describes the likelihood of a given number of events occurring in a fixed interval of time or space. It is characterized by a single parameter, usually denoted by ? (lambda), which represents the average rate of occurrence. This distribution is particularly useful for modeling count data when events occur independently and at a constant average rate.

Parameter Scaling in Poisson Processes

In the context of a Poisson process, the parameter ? needs to be scaled appropriately when considering different time intervals. The average rate ? given for a specific time unit can be adjusted by multiplying it by the length of the time interval in question. This scaling is crucial for accurately computing probabilities over any interval that is different from the base interval.

Transcript

00:01 So traffic flow is traditionally modeled by poisson distribution.

00:06 So we are going to be using poisson probabilities here, which your formula would be the probability that x is some number, we're going to call it x, equals e to the negative lambda times lambda to the x power over x factorial.

00:29 So we're provided a mean in this problem, and it tells us that we have an average of six cars in one minute.

00:41 So in part a, we're going to have to first scale that down because we're changing our time frame to be 30 seconds.

00:50 So if there are six cars in one minute, which is equivalent to 60 seconds, then how many cars are there in 30 seconds? and if you were to cross multiply or if you were to write an equivalent fraction, you would find that lambda is going to equal three.

01:11 So in part a, we want the probability that there are no cars.

01:16 So this is, x is going to be zero.

01:19 So we'll apply our formula.

01:22 It's e to the negative lambda power multiplied by lambda to the x, and we said x was zero, over zero.

01:32 Factorial.

01:33 And in your algebra days, three to the zero power is the same as one.

01:38 Zero factorial is the same as one.

01:40 So it's really equivalent to e to the negative three power.

01:44 And if you were to pick up your calculator, you would calculate e to the negative three power as 0 .049 -887.

01:57 And depending on your professor or your teacher and how many places they prefer you to round your answer to.

02:05 I usually have my classes round to four decimal places, so 0 .498.

02:13 In part b, we are using the same time frame, so therefore we're going to have the same lambda value.

02:23 But this time we want the probability that we have three or more cars.

02:28 So now we're talking x is greater than are equal to three.

02:33 So pusan probabilities are discrete probabilities.

02:38 So that means we have to think in terms of a table of values.

02:44 So we could have had no cars in the intersection, or one, or two, or three, or four, or five, or six, or seven.

02:57 And that table is never going to end.

03:00 So with discrete probability distributions, you do have to realize that the sum of all the probabilities will always equal one.

03:13 So if we think of the fact that this chart can be broken into two sections, we can talk about less than or equal to two and greater than or equal to three, then we could say that the probability that x is less than or equal to two, plus the probability that x is greater than or equal to three has to equal one.

03:36 And if we subtract the probability that x is less than or equal to two from both sides, then we have that the probability that x is greater than or equal to three is equivalent to one minus the probability that x is less than or equal to two.

03:58 So we're going to utilize our formula multiple times.

04:04 So we already have the value when x is zero.

04:07 We know that that is 0 .04, 97, 8, 7, 06 ,87.

04:17 That was from part a.

04:19 If we replace the x in our formula with a 1, which means we're going to change this exponent to a 1 and this denominator to a 1 factorial, we will get that the probability that x equals one or that there's one car in a 30 second interval to be 0 .149361 -2051 if x is 2 so we'll change this to be a 2 and this to be a 2 you will get 0 .2404 .1.

05:03 8807.

05:04 So therefore, the probability that x is less than or equal to two will be these totaled up.

05:12 And if i total those up, i'm going to get a value of 0 .4 -2 -319 -0 -811.

05:23 So when i subtract that from 1, i'm getting 0 .5768099 -189 .9 .9.

05:32 And again, we're going to round to four decimal places, so we would say there is a probability of 0 .5768 that three or more cars pass through that intersection within 30 seconds.

05:50 For part c, we're going to want to extend our chart a little bit...

Please give Ace some feedback