In Exercises 9–16, use the Poisson distribution to find the indic...

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

Statistics

5. Binomial Distribution & Discrete Random Variables

Discrete Random Variables

1:35 minutes

Problem 5.3.9a

Textbook Question

In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.

Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.

a. Find the mean number of births per day.

Verified step by step guidance

Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the number of events (e.g., births) occurring in a fixed interval of time or space, given the events occur independently and at a constant average rate.

Step 2: Identify the total number of events and the time interval. In this problem, there are 5942 births over 365 days.

Step 3: Calculate the mean number of births per day. The mean (λ) for a Poisson distribution is the total number of events divided by the total time interval. Use the formula:

λ = \frac{5942}{365}

Step 4: Simplify the fraction to find the mean number of births per day. This value represents the average rate of births per day at the medical center.

Step 5: Use this mean (λ) for further calculations if needed, such as finding probabilities of specific numbers of births per day using the Poisson probability formula.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Poisson Distribution

The Poisson distribution is a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur with a known constant mean rate and independently of the time since the last event. It is particularly useful for modeling rare events, such as the number of births in a day.

Mean of a Poisson Distribution

In a Poisson distribution, the mean (λ) represents the average number of occurrences of the event in the specified interval. For the problem at hand, to find the mean number of births per day, you would divide the total number of births by the number of days in the year, providing a clear understanding of the expected daily occurrences.

Expected Value

The expected value is a key concept in probability that provides a measure of the center of a probability distribution. In the context of the Poisson distribution, the expected value is equal to the mean (λ), which indicates the average outcome one can anticipate over a specified period. This concept helps in making predictions about future occurrences based on historical data.

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In Exercises 9–16, use the Poisson distribution to find the indicated probabilities.Births In a recent year (365 days), NYU-Langone Medical Center had 5942 births.a. Find the mean number of births per day.