The number of failures of a testing instrument from...

Name: Problem 168, Chapter 3: Applied Statistics and Probability for Engineers
Uploaded: 2021-08-11T13:52:30-08:00
Duration: 6 min 20 s
Channel: Sonam Khatri
Description: The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failures per hour. (a) What is the probability that the instrument does not fail in an 8 -hour shift? (b) What is the probability of at least one failure in a 24 -hour day?

The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failures per hour. (a) What is the probability that the instrument does not fail in an 8 -hour shift? (b) What is the probability of at least one failure in a 24 -hour day?

The number of failures of a testing instrument from contamination particles on the product is a Poisson random variable with a mean of 0.02 failures per hour.
(a) What is the probability that the instrument does not fail in an 8 -hour shift?
(b) What is the probability of at least one failure in a 24 -hour day?

Key Concepts

Probability Mass Function (PMF)

The PMF of a Poisson distribution provides the probability of observing exactly k events in a given interval. It is defined by the formula P(X = k) = (?t)^k * exp(??t) / k!, where ? is the rate parameter and t is the length of the interval.

Time Interval Scaling

In problems involving Poisson processes, the mean number of events is proportional to the length of the time interval. Thus, if the rate is given per hour, multiplying the rate by the number of hours yields the expected number of events over that interval, which is then used in the PMF calculation.

Poisson Distribution

This is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, assuming that these events occur with a known constant mean rate and independently of the time since the last event.

Transcript

00:04 In this problem, it is given that the number of failures of a testing instrument is a poison random variable with a mean of 0 .02 failures per hour.

00:16 Let x denote the number of failures of testing instrument in one hour.

00:23 So, lembra is 0 .02 per hour.

00:30 We know that if x is a poison random variable, then probability of x equal to.

00:36 To x is e raised to minus lambda t into lambda t raised to x divided by x factorial where x ranges from 0 1 to up to infinity in the first part it is asked what is the probability that the instrument does not fail in an 8 hour shift so here t is equal to 8 as the probability that the instrument does not fail is asked for an 8 hour shift here t is equal to 8.

01:22 So lembra t is equal to 0 .02 into 8 is 0 .16.

01:36 Probability that the instrument does not fail is probability of x equal to 0.

01:44 That is no failure occurs.

01:49 This is equal to e raised to much.

01:52 Minus lambda t but lambra t is 0 .16 so e raised to minus 0 .16 into 0 .16 raised to x that is 0 .16 raised to 0 divided by x factorial that is 0 factorial this is equal to e raised to minus 0 .16 .0 .1 16 raised to 0 is 1 and 0 factorial is also 1.

02:32 So e raised 2 minus 0 .16 into 1 divided by 1 which is equal to e raised 2 minus 0 .16.

02:44 This is equal to 0 .8521.

02:56 So the probability that the instrument does not fail in an 8 hour shift is 0 .85 to 1.

03:04 To 1...

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