Suppose small aircraft arrive at an airport according to a Poisson process with rate λ=8 per hou

Suppose small aircraft arrive at an airport according to a...

Key Concepts

Poisson Process

A Poisson process models random events occurring independently and at a constant average rate over time. It is widely used to represent counts of events in fixed time intervals, such as arrivals or occurrences, where the probability of a single event occurring in a small interval is proportional to the length of the interval.

Poisson Distribution

The Poisson distribution is the probability distribution of the number of events in a fixed interval of time or space when these events occur with a known constant mean rate and independently of the time since the last event. Its probability mass function is P(X = k) = e^(??) ?^k/k!, where ? is the expected number of events in the interval.

Parameter Scaling With Time

In a Poisson process with a constant rate ?, the parameter ? for the Poisson distribution is scaled by the length of the time interval (t), such that ? = ?t. This scaling allows the model to be applied to intervals of different durations, maintaining the property that the mean and variance of the distribution are both equal to ?.

Mean and Variance of the Poisson Distribution

For a Poisson random variable, both the mean and the variance equal the parameter ? (the product of the arrival rate and time interval). This implies that the standard deviation is the square root of ?. These properties facilitate the calculation of measures of central tendency and dispersion for events modeled by the Poisson process.

Cumulative Probability Calculations

Cumulative probabilities for a Poisson random variable can be calculated by summing the probability mass function over the desired range of values, or by using the complement rule. For example, the probability of at least k events is often computed as one minus the probability of fewer than k events, both of which are essential techniques for handling 'at least' or 'at most' type probability questions.

Transcript

00:01 Okay, so with this problem, we have a passan process with the rate of eight per hour so that the parameter is m equals 18.

00:13 And the first part of this question is asking us to find the probability of six ships arriving in one hour.

00:20 And then we have to find the probability of at least six and then the probability of at least ten.

00:24 So to do that, we've got our first part here.

00:30 So we got the probability number of ships arriving.

00:34 Equals six.

00:37 All right.

00:37 And to do that, we got the probability six, eight.

00:45 We need to do this, we've got, just to plug it into the equation.

00:52 We have e.

00:53 The negative eighth.

00:55 Sorry, i'll make that a little clear.

00:57 Equal negative eighth times eight to the sixth.

01:09 Eight to the sixth.

01:19 Sorry, there we go.

01:25 Eight to the six.

01:26 Over six sorry about that my screen is acting weird for a second and then this answer comes out to be 0 .1 so that's your answer to the first part of part a the next part asks us to find the probability of at least six ships arriving so to do that to find the probability that at least six will arrive we've got one minus probability of less than six arriving.

02:06 All right.

02:07 And that makes sense because we can easily find the probability of less than six, but we will have no way of calculating the way of more than six without taking this initial step right here.

02:16 Okay.

02:17 Let's do this.

02:20 We've got one minus summation of zero to five to a probability and eight.

02:36 And then this equals 1 minus summation of e to the negative eighth times 8 to the x over x factorial.

02:57 And this math comes out to be 0 .81.

03:03 So the hardest part about this is first making this initial step.

03:07 That's a key step right there.

03:09 And then you also just have to plug numbers into the equation.

03:12 Right this eight in case you're wondering comes from the initial parameters that we were given and that is i'll do it over here in blue i'll do it over here in yeah we've got m equals 8t and since t equals 1 we got m equals 8 and that's where the 8 that you're seeing is coming from and all of these equations all right so that's your answer to the second part and then the last part of part part a is asking us to find the probability that at least 10 ships arrive during this time period.

03:56 Right, and we're going to take the same step that we did in part one, which is going to do 1 minus the probability that less than 10 ships arrive for the same reason, because we cannot possibly calculate the number between 10 and infinity.

04:11 So this looks like a very similar problem.

04:14 1 minus the summation 0 to 9, right? 0 to 9, we don't include 10 because we're going less than then.

04:25 Okay, and this equals 1 minus summation from 0 to 9 of of e to the negative 8th times 8 to the x over x.

04:50 Right, so if you look between this, problem and the one in part b the only thing that changes is the parameters on a summation right 0509 other than that they're the same problem sorry about that all right and then this part sorry this part comes out to 0 .283 okay so that's part a for you we'll scroll down we'll tap into part b part b is pretty self -explanatory.

05:31 Part b is asking you to find the expected value and standard deviation of ships arriving in a 90 -minute time period.

05:39 Right.

05:40 And to do this, you still need, you need to use that same equation that we had written up there in blue.

05:44 I'll write it again.

05:47 That equation is m equals 8t.

05:51 And since you've got a 90 -minute window, you've got 90 minutes times 60 minutes, no, sorry, 90 minutes times one hour over 60 minutes...

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