The receptionist of JVMRG Hotel received an average of 5...

Name: Problem 11, Chapter 3: Statistics
Uploaded: 2024-05-13T03:14:27-08:00
Duration: 4 min 21 s
Channel: Prabhakar Kumar
Description: The receptionist of JVMRG Hotel received an average of 5 phone calls every hour. What is the probability? a. that on the next hour she will receive exactly 6 phone calls b. that on the next hour she will receive at most 10 phone calls c. that on the next hour she will receive from 7 to 9 phone calls d. that on the next hour she will receive at least 8 phone calls

The receptionist of JVMRG Hotel received an average of 5 phone calls every hour. What is the probability? a. that on the next hour she will receive exactly 6 phone calls b. that on the next hour she will receive at most 10 phone calls c. that on the next hour she will receive from 7 to 9 phone calls d. that on the next hour she will receive at least 8 phone calls

The receptionist of JVMRG Hotel received an average of 5 phone calls every hour. What is the probability?
a. that on the next hour she will receive exactly 6 phone calls
b. that on the next hour she will receive at most 10 phone calls
c. that on the next hour she will receive from 7 to 9 phone calls
d. that on the next hour she will receive at least 8 phone calls

Key Concepts

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, provided these events occur with a known constant mean rate and independently of the time since the last event.

Probability Mass Function (PMF)

The probability mass function is used to calculate the probability of observing exactly a certain number of events in a Poisson distribution. It is given by the formula P(X = k) = (e^(-?) * ?^k) / k!, where ? is the average rate and k is the number of occurrences.

Cumulative Distribution Function (CDF)

The cumulative distribution function in the context of a Poisson distribution provides the probability that the random variable is less than or equal to a specified value. This is useful for determining probabilities for ranges, such as at most a certain number of events or between two values.

Parameter (?) in the Poisson Process

The parameter ? represents the average rate of occurrence of events per interval in a Poisson process. It is a critical value used within the PMF and CDF formulas to determine the likelihood of different outcomes within the specified interval.

Transcript

00:01 In this question, the motive is to calculate the probability values such that the random variable x which represents the number of phone calls will follow poisson distribution because number of events are happening in a fixed interval of time.

00:17 So, that's why it will follow here poison distribution with the parameter value of 5 and then we can say that the probability mass function for poison distribution can be here written as probability of x is equal to k and then it is e raised to the power negative lambda lambda raised to the power k divided by it will be k factorial.

00:42 So, in the first part of this very problem let us write it is part a and then it will be probability of x is equals to 6.

00:52 So, this is here equals to e raised to the power negative 5 and then it is 5 raised to the power 6 divided by 6 factorial which is here equals to 0 .14 and then it will be 622 and we can put this very answer inside the box in order to highlight it.

01:13 Now let us discuss the second part of this very problem.

01:17 So here it will be probability of receiving at most 10 phone calls in the next hour.

01:24 So it is probability of x less than equals to 10 and it can be here written as it will be summation and then it is e raised to the power negative 5, 5 raised to the power k and then in the denominator it is k factorial right.

01:42 So let us write over here k factorial and the value of k will vary from 0 to 10 because we need to find the probability of x less than equals to 10.

01:53 So it will be here equals to 0 .98 and it is 63 so we can put this very answer inside the box in order to highlight it.

02:03 Now let us see how we can calculate the probability in the third part of this very problem.

02:09 So it will be probability of 7 less than equals to x and that is less than equals to 9...

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