The number of views of a page on a Web site follows a...

Name: Problem 142, Chapter 3: Applied Statistics and Probability for Engineers
Uploaded: 2021-09-08T03:18:18-08:00
Duration: 6 min 18 s
Channel: Evelyn Cunningham
Description: The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute. (a) What is the probability of no views in a minute? (b) What is the probability of two or fewer views in 10 minutes? (c) Does the answer to the previous part depend on whether the 10 -minute period is an uninterrupted interval? Explain.

The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute. (a) What is the probability of no views in a minute? (b) What is the probability of two or fewer views in 10 minutes? (c) Does the answer to the previous part depend on whether the 10 -minute period is an uninterrupted interval? Explain.

The number of views of a page on a Web site follows a Poisson distribution with a mean of 1.5 per minute.
(a) What is the probability of no views in a minute?
(b) What is the probability of two or fewer views in 10 minutes?
(c) Does the answer to the previous part depend on whether the 10 -minute period is an uninterrupted interval? Explain.

Key Concepts

Time Scaling in Poisson Processes

Time scaling in Poisson processes refers to the property that the number of events in a longer interval can be modeled as a Poisson distribution with a rate equal to the original rate multiplied by the length of the interval. This property simplifies calculation over different time periods, as the overall distribution over any interval is still Poisson, with its mean adjusted for the length of the interval.

Independence of Increments

The independence of increments is a key concept in Poisson processes, which posits that the number of events occurring in non-overlapping intervals are independent of each other. This property implies that splitting an interval into several segments does not affect the distribution of events in the overall interval, thus ensuring that computations remain consistent regardless of whether the time period is viewed as a single continuous interval or partitioned into separate parts.

Poisson Distribution

The Poisson distribution is a probability distribution that describes the number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It is widely used to model count-based phenomena in various fields such as physics, finance, and traffic engineering.

Probability Mass Function

The probability mass function (PMF) for a Poisson random variable gives the probability that the variable takes a specific discrete value. It is expressed as P(X = k) = (?^k e^(-?)) / k! where ? is the average number of events in the interval, k is the number of events, and e is the base of the natural logarithm. This function is crucial for computing the likelihood of any given number of events within the specified period.

Transcript

00:01 So we are dealing with a poisson distribution.

00:04 So we want to talk about poisson probability.

00:11 And the formula for prasan probability is the probability that x equals some number, i'll call little x, is equal to e to the negative lambda times lambda to that x power all over x factorial.

00:29 So we were given a mean, and the mean we were given was that there were 1 .5.

00:35 Views in one minute and in part a we are trying to find the probability that there are no views and the time frame is one minute so therefore lambda is going to be the 1 .5 views so we'll apply our formula e to the negative lambda times lambda to the x power which we decided or declared x to be zero over zero factorial.

01:12 And from your algebra days, anything to the zero power is equivalent to one.

01:17 Zero factorial is equivalent to one.

01:20 So in essence, it's just e to the negative 1 .5 power, which is equivalent to 0 .22311601.

01:31 And we tend to round our probabilities to four decimal places, so it would be 0 .223 .1.

01:43 When we look at part b, now we are switching our time frame to 10 minutes.

01:52 So if it was 1 .5 views in one minute, that would be equivalent to how many views in 10 minutes.

02:03 And if we were to write our equivalent fraction, we could cross multiply and we would get that lambda equals 15 views.

02:14 So in this problem, we want to determine the probability of two or fewer.

02:22 So that could be written as x is less than or equal to two...

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