5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
3:44 minutesProblem 5.3.9c
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.
c. Find the probability that in a single day, there are no births. Would 0 births in a single day be a significantly low number of births?
Verified step by step guidance1
Step 1: Understand the Poisson distribution. The Poisson distribution is used to model the probability of a given number of events occurring in a fixed interval of time or space, given a known average rate of occurrence (λ). The formula for the Poisson probability is P(X = k) = (λ^k * e^(-λ)) / k!, where k is the number of events, λ is the average rate, and e is the base of the natural logarithm (approximately 2.718).
Step 2: Calculate the average number of births per day (λ). Since there were 5942 births in 365 days, divide the total number of births by the number of days to find the daily average: λ = 5942 / 365.
Step 3: Use the Poisson formula to calculate the probability of no births in a single day (k = 0). Substitute k = 0 and the calculated value of λ into the formula: P(X = 0) = (λ^0 * e^(-λ)) / 0!. Note that 0! = 1 and λ^0 = 1, so the formula simplifies to P(X = 0) = e^(-λ).
Step 4: Determine whether 0 births in a single day is a significantly low number. A result is typically considered significantly low if its probability is less than or equal to 0.05. Compare the calculated probability P(X = 0) to this threshold to make a conclusion.
Step 5: Interpret the result. If P(X = 0) is less than or equal to 0.05, then 0 births in a single day would be considered significantly low. Otherwise, it would not be considered significantly low. Provide reasoning based on the calculated probability and the context of the problem.
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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 happen 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.
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Mean (λ) in Poisson Distribution
In the context of the Poisson distribution, the mean (denoted as λ, lambda) represents the average number of occurrences of the event in the specified interval. For the given problem, λ would be calculated by dividing the total number of births (5942) by the number of days (365), which provides the expected number of births per day.
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Significance of Events
Determining whether an event is significantly low involves comparing the observed outcome to the expected outcome under the Poisson distribution. In this case, if the probability of observing 0 births in a day is very low (typically below a threshold like 0.05), it may be considered significantly low, indicating that such an occurrence is unusual given the average rate of births.
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