The number of telephone calls that arrive at a phone...

Key Concepts

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 the events occur independently and at a constant average rate. This distribution is widely used in fields such as telecommunications, astronomy, and particle physics, where it models random counts of events like telephone calls, arrivals, or decays.

Probability Mass Function (PMF)

The probability mass function for the Poisson distribution gives the probability of exactly k events occurring in a fixed interval. It is defined by the formula P(X=k) = (e^(-?) * ?^k) / k!, where ? is the expected number of events in the interval and k is a non-negative integer. This formula is the key tool for calculating the probabilities for various specific outcomes.

Mean Rate (?) and Scaling with Time

The parameter ? in the Poisson distribution represents both the mean and the variance of the distribution, indicating the average number of events occurring in a given interval. When adjusting for different time intervals, ? scales proportionally to the length of the interval. For instance, if the average rate is given per hour, then for a half-hour period the rate becomes half of the hourly rate. This scaling property is essential in applying the Poisson model to various scenarios.

Cumulative Distribution Function (CDF)

The cumulative distribution function for the Poisson distribution represents the probability that the random variable is less than or equal to a certain value. It is obtained by summing the probabilities of all individual outcomes up to that value using the PMF. The CDF is particularly useful when determining the likelihood of 'three or fewer' events, as it aggregates the probabilities of several outcomes into a single measure.

Transcript

00:03 In this problem, it is given that the number of telephone calls that arrive at a phone exchange is often modeled as a poison random variable.

00:13 Assume that on the average, there are 10 calls per hour.

00:19 Let x denote the number of telephone calls that arrive at the phone exchange per hour.

00:27 It is said on the average there are 10 calls per hour.

00:32 So here, lambrata t is equal to 10 in 10.

00:36 To 1 which is equal to 10.

00:40 We know that probability of x equal to x for a poison random variable is e raised to minus lambda t, lambda t raised to x divided by x factorial where x ranges from 0 1 to up to infinity.

00:58 So this is equal to e raised to minus 10, 10 raised to x divided by x factorial.

01:18 As here, lambda t is equal to 10.

01:21 So this is equal to e raised to minus 10, 10 raised to x divided by x factorial.

01:30 In the first part, beer asked, what is the probability that there are exactly five calls in one hour.

01:40 That is, we have to find probability of x equal to 5.

01:48 Probability of x equal to 5 is e raised to minus 10 multiplied by 10 raised to 5 divided by 5 factorial.

02:08 This is equal to 0 .0378.

02:14 0 .0 .78.

02:22 0 .0378.

02:24 So the probability that there are exactly 5 calls in 1 hour is 0 .0378.

02:34 Next, it is asked, what is the probability that there are 3 or fewer calls in 1 hour? that is, we have to find probability of x less than or equal to 3.

02:53 This is equal to, as the poison random variable starts from x equal to 0, we will start from probability of x equal to 0.

03:06 So this is equal to probability of x equal to 0 plus probability of x equal to 1 plus probability of x equal to 2 plus probability of x equal to 3.

03:39 As we have to find probability of x less than or equal to 3 we will add the probabilities from x equal to 0 up to x equal to 3 this is equal to e raised 2 minus 10 10 raised to 0 divided by 0 factorial plus e raised to minus 10 10 raised to 1 now we are finding probability of x equal to 1 so 10 raised to 1 divided by 1 factorial plus e raised to minus 10, 10 raised to 2.

04:42 Now we are finding probability of x equal to 2.

04:45 So 10 raised to 2 divided by 2 factorial plus e raised to minus 10, 10 raised to 3 divided by 3 factorial.

05:08 Now we will find these values.

05:11 This is equal to 0 .005 plus 0 .0045 plus 0 .0045 plus 0 .00227 plus 0 .007 plus 0 .0075.

06:04 7.

06:08 This is equal to this sum is equal to 0 .0103.

06:18 0 .0103.

06:22 So the probability that there are three or fewer calls in one hour is 0 .0103.

06:32 Next it is asked what is the probability that there are exactly 15...