The number of messages that arrive at a Web site is a...

Name: Problem 190, Chapter 3: Applied Statistics and Probability for Engineers
Uploaded: 2021-08-14T09:03:39-08:00
Duration: 6 min 55 s
Channel: Sonam Khatri
Description: The number of messages that arrive at a Web site is a Poisson random variable with a mean of five messages per hour. (a) What is the probability that five messages are received in 1.0 hour? (b) What is the probability that 10 messages are received in 1.5 hours? (c) What is the probability that fewer than two messages are received in 0.5 hour?

The number of messages that arrive at a Web site is a Poisson random variable with a mean of five messages per hour. (a) What is the probability that five messages are received in 1.0 hour? (b) What is the probability that 10 messages are received in 1.5 hours? (c) What is the probability that fewer than two messages are received in 0.5 hour?

The number of messages that arrive at a Web site is a Poisson random variable with a mean of five messages per hour.
(a) What is the probability that five messages are received in 1.0 hour?
(b) What is the probability that 10 messages are received in 1.5 hours?
(c) What is the probability that fewer than two messages are received in 0.5 hour?

Key Concepts

Rate Scaling in Poisson Processes

In a Poisson process, the mean number of events (?) scales linearly with the length of the time interval if the rate of events is constant. This means that for time intervals of different lengths, the expected number of events can be adjusted by multiplying the basic rate by the length of the interval. This adjustment is crucial for correctly applying the Poisson distribution to various time spans.

Probability Mass Function

The probability mass function (PMF) for the Poisson distribution is used to calculate the probability of a given number of events occurring in a fixed interval. The PMF is defined as P(X = n) = (e^(-?) * ?^n) / n!, where ? represents the expected number of events and n is the number of events. This formula is essential for determining the likelihood of observing a specified count of events.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, under the assumption that these events occur with a known constant mean rate and independently of the time since the last event. It is widely used for modeling count data that represents the number of times an event happens within a given period.

Transcript

00:03 In this problem it is given that the number of messages is a poison random variable with a mean of 5 messages per hour.

00:14 Let x denote the number of messages received per hour.

00:19 So lambda is equal to 5 as the mean is 5 messages per hour.

00:26 So lambda is 5.

00:29 We know that if x is a poison random variable, then probability of x equal to x is is e raised to minus lambda t into lambda t raised to x divided by x factorial, where x goes from 0, 1, 2 up to infinity.

00:51 In the first part, it is asked what is the probability that 5 messages are received in 1 hour? for 1 hour, t is equal to 1.

01:06 So, lambda t is equal to 5 into 1, which is equal.

01:11 To 5.

01:16 Probability that 5 messages are received.

01:19 That is, we have to find probability of x equal to 5.

01:28 This is equal to e raised to minus lambda t, that is, e raised to minus 5, lambda t raised to x, that is 5 raised to 5, divided by x factorial so divided by five factorial e raised to minus five into five into five raised to five divided by five factorial this is equal to zero point one seven five five so the probability that five messages are received in one hour is zero point one seven five five in the next part it is asked, what is the probability that 10 messages are received in 1 .5 hours? for 1 .5 hours, t is equal to 1 .5.

02:31 So, lambda t is equal to 5 into 1 .5, which is equal to 7 .5.

02:41 Lembera t is 7 .5.

02:46 Probability that 10 messages are received.

02:50 So we have to find probability of x equal to 10.

02:58 This is equal to e raised to minus lambda t, that is e raised to minus 7 .5 into lambda t raised to x, that is 7 .5 raised to 10 divided by x factorial, that is divided by 10 factorial...

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