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Chapter 3: Problem 207

The number of people entering the intensive care unit at a hospital on any single day possesses a Poisson distribution with a mean equal to five persons per day. a. What is the probability that the number of people entering the intensive care unit on a particular day is equal to 2 ? Is less than or equal to 2 ? b. Is it likely that \(Y\) will exceed \(10 ?\) Explain.

Short Answer

Expert verified
Probability exactly 2 is 0.0842; \( \leq 2 \) is 0.1247. Unlikely \( Y > 10 \).

Step by step solution

01

Identify the Distribution

The problem states that the number of people entering the intensive care unit follows a Poisson distribution with a mean of 5 persons per day. Hence, we have a Poisson distribution with parameter \( \lambda = 5 \).
02

Calculate Probability for Exactly 2 People

For a Poisson distributed random variable \( X \), the probability of observing \( x \) events in an interval is given by \( P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \). For 2 people entering, \( x = 2 \), \( \lambda = 5 \):\[P(X = 2) = \frac{e^{-5} \times 5^2}{2!} = \frac{e^{-5} \times 25}{2}\]Calculate this using a calculator.
03

Calculate Probability for 0, 1, and 2 People

To find the probability of \( X \leq 2 \), calculate \( P(X = 0) \), \( P(X = 1) \), and \( P(X = 2) \), then sum these probabilities. Use the Poisson formula:\[P(X = 0) = \frac{e^{-5} \times 5^0}{0!} = e^{-5}\]\[P(X = 1) = \frac{e^{-5} \times 5^1}{1!} = 5e^{-5}\]Combine with \( P(X = 2) \) from Step 2.
04

Assess Likelihood of Exceeding 10 People

To determine if \( Y > 10 \) is likely, calculate \( P(X \leq 10) \) using a cumulative Poisson distribution table or software, then use \( P(X > 10) = 1 - P(X \leq 10) \). If \( P(X > 10) \) is very small, it is unlikely \( Y \) will exceed 10.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Intensive Care Unit
An Intensive Care Unit (ICU) is a specialized section in a hospital where patients receive intense and constant care. This is usually due to life-threatening conditions or surgeries.
The staffing in ICUs is unique, with highly trained nurses, doctors, and other professionals who are adept at handling intricate and emergency situations.
  • ICUs have advanced technology and medical equipment.
  • The patient-to-staff ratio is low for more personalized care.
  • Only severe or critical patients are admitted.
In statistical modeling, like using the Poisson distribution, the focus is often on events such as the number of people entering an ICU each day.
This helps manage resources, ensuring that patients are given the immediate care they need. Modeling such events allows hospital administrators to better plan and allocate manpower for efficient and effective healthcare delivery.
Probability Calculation
Probability calculation is essential in understanding events that have inherent uncertainty. This can include how many patients will enter an ICU on any given day.
For a Poisson distribution, we have a specific formula used to calculate probabilities. It is designed to model the number of times an event occurs within a fixed interval of time or space.
The probability of having exactly two patients, for example, can be calculated using the Poisson formula: \[ P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!} \] Where:
  • \( x \) is the number of occurrences we want to calculate the probability for (such as 2 patients entering the ICU).
  • \( \lambda \) is the average rate of occurrences (mean) over the interval (5 patients per day here).
  • \( e \) is the base of the natural logarithm (approximately 2.71828).
These probability calculations allow hospitals to predict patient inflow, thus helping them to manage staffing and resources effectively.
Statistical Modeling
Statistical modeling involves using mathematical frameworks to represent real-world situations. It's crucial for predicting outcomes and trends based on historical data. In healthcare, like monitoring ICU admissions, statistical modeling helps in decision-making.
In this context, the Poisson distribution is a powerful model. It's particularly used when we need to predict the number of events in a given timeframe, assuming these events happen independently over a continuous interval.
  • Models help in understanding potential fluctuations in patient inflow.
  • Aids in strategic resource allocation, like medical staff and equipment.
  • Helps in assessing the probability of rare occurrences (like having more than a specific number of patients in one day).
By assessing probabilities, like if over 10 patients might enter an ICU on a particular day, hospitals can prepare contingency plans. This ensures that care is always available, even during unexpected situations.

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Most popular questions from this chapter

In southern California, a growing number of individuals pursuing teaching credentials are choosing paid internships over traditional student teaching programs. A group of eight candidates for three local teaching positions consisted of five who had enrolled in paid internships and three who enrolled in traditional student teaching programs. All eight candidates appear to be equally qualified, so three are randomly selected to fill the open positions. Let \(Y\) be the number of internship trained candidates who are hired. a. Does \(Y\) have a binomial or hypergeometric distribution? Why? b. Find the probability that two or more internship trained candidates are hired. c. What are the mean and standard deviation of \(Y\) ?

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