Customers arrive at a checkout counter in a department store...

Name: Problem 122, Chapter 3: Mathematical Statistics with Applications
Uploaded: 2023-06-14T00:57:49-08:00
Duration: 7 min 21 s
Channel: Aparna Shakti
Description: Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that a. no more than three customers arrive? b. at least two customers arrive? c. exactly five customers arrive?

Customers arrive at a checkout counter in a department store according to a Poisson distribution at an average of seven per hour. During a given hour, what are the probabilities that a. no more than three customers arrive? b. at least two customers arrive? c. exactly five customers arrive?

Customers arrive at a checkout counter in a department store according to a Poisson distribution
at an average of seven per hour. During a given hour, what are the probabilities that
a. no more than three customers arrive?
b. at least two customers arrive?
c. exactly five customers arrive?

Key Concepts

Complement Rule

The complement rule in probability states that the probability of an event occurring is equal to one minus the probability of it not occurring. This rule is particularly useful for finding the probability of 'at least' a certain number of events by subtracting the probability of having fewer events from one. It simplifies the computation especially when direct calculation of the desired probability is complex.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring within a fixed interval of time or space, provided that these events occur with a known constant mean rate and independently of the time since the last event. It is commonly used in scenarios where events happen at a random and independent manner over continuous time or space.

Probability Mass Function (PMF)

The probability mass function for the Poisson distribution quantifies the probability of exactly k events occurring in a fixed interval. The function is given by P(X = k) = (?^k * e^(-?)) / k!, where ? is the average number of events in the interval, k is the number of occurrences, and e is the base of the natural logarithm. This formula is essential for calculating the probabilities of specific event counts.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) for the Poisson distribution sums the probabilities of observing up to a certain number of events. It is used to compute the probability that the random variable does not exceed a specified value, such as finding the probability of observing no more than a certain number of occurrences. This is particularly useful in calculating cumulative probabilities in real-world contexts.

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