Question

The number of births at the local hospital has a Poisson distribution with an average of 6 per day. a. What is the probability distribution for the daily number of births at this hospital? b. What is the probability distribution for the number of hourly births? c. What is the probability that there are fewer than 3 births in a given hour? d. Within what interval would you expect to find the number of hourly births at least \(89 \%\) of the time?

Short answer

Answer: The interval for hourly births containing 89% probability is [0, 1].

Step-by-step solution

Understanding Poisson Distribution

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate (λ, lambda) and are independent of the time since the last event. The formula for Poisson distribution is: P(X = k) = (e^(-λ) * (λ^k)) / k! where - P(X = k) is the probability of observing exactly k events, - λ is the average rate (number of events per interval), - e is the base of the natural logarithm (approximately 2.71828), - k is the number of events. #Step 2: Daily probability distribution#

Daily Probability Distribution

We are given the average number of births per day as 6. So, λ for the daily Poisson distribution is 6. Thus, the probability distribution for the daily number of births at this hospital is: P(X = k) = (e^(-6) * (6^k)) / k! #Step 3: Hourly probability distribution#

Hourly Probability Distribution

We are asked to find the probability distribution for hourly births. Since there are 24 hours in a day, we can calculate the average number of births per hour by dividing the daily average by 24: Hourly λ = 6 / 24 = 0.25 Thus, the probability distribution for the number of hourly births at this hospital is: P(X = k) = (e^(-0.25) * (0.25^k)) / k! #Step 4: Probability of fewer than 3 births per hour#

Probability of Fewer Than 3 Births per Hour

We will now use the hourly probability distribution to find the probability of fewer than 3 births in a given hour. We will calculate it by summing the probabilities of 0, 1, and 2 births using the formula: P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2) = (e^(-0.25) * (0.25^0)) / 0! + (e^(-0.25) * (0.25^1)) / 1! + (e^(-0.25) * (0.25^2)) / 2! = e^(-0.25) * (1 + 0.25 + (0.25^2) / 2) Approximately, P(X < 3) = 0.9692 #Step 5: Interval for hourly births at least 89% of the time#

Finding the Interval

To find the interval within which the number of hourly births occurs at least 89% of the time, we will find the cumulative probabilities for each number of births, starting from 0, until it exceeds 0.89. We will use the hourly Poisson distribution formula: Cumulative probability = P(X = 0) + P(X = 1) + ... + P(X = k) Let k be the upper limit of the interval, we need to find the smallest k while the cumulative probability > 0.89. By calculating the probabilities for different k values, we find that: P(X=0) ≈ 0.7788 P(X=0 or 1) ≈ 0.9662 P(X=0 or 1 or 2) ≈ 0.9970 Since the cumulative probability exceeds 0.89 between 0 and 1 births, the interval is [0, 1]. Therefore, the number of hourly births at least 89% of the time will be within the interval [0, 1].

Key concepts

Understanding Probability Distributions

Probability distributions are mathematical functions that describe the likelihood of different outcomes in a random experiment. A probability distribution assigns a probability to each outcome in the sample space of a random event. If you roll a die, for example, the probability distribution would show an equal likelihood for each number from 1 to 6 since it's a fair six-sided die.

When discussing probability distributions, there are two main categories: continuous and discrete. Discrete probability distributions, like the Poisson distribution mentioned in the exercise, list the probabilities of outcomes that are countable and finite, like the number of births at a hospital. Meanwhile, a continuous distribution would deal with an infinite number of possible outcomes within a range, such as weights or heights.

The Poisson distribution specifically deals with the probability of a given number of events happening within a fixed interval of time or space, given a constant average rate of occurrence. It's discrete because you can count the number of events, like births, and it applies to scenarios where events happen independently of each other.

Exploring Discrete Probability

Characteristics of Discrete Probability Distributions

In discrete probability distributions, outcomes are distinct and countable. The characteristics that define these distributions include:

• Being based on a countable number of distinct outcomes.
• Probability of each outcome must be between 0 and 1, inclusive.
• The sum of the probabilities of all possible outcomes equals 1.

In our exercise, considering the daily and hourly births at a local hospital is a classic example of such a distribution, where the Poisson distribution helps calculate the probability of observing a specific number of births.

The formula used to calculate these probabilities is designed to take into account the average rate of occurrence (denoted by lambda, \(\lambda\)) and, applying the formula can yield the probability of having exactly k events in the given time frame. This granular level of prediction is particularly useful in healthcare, insurance, and other fields that rely on quantifying the likelihood of discrete events.

Cumulative Probability in Discrete Distributions

What is Cumulative Probability?

Cumulative probability is the sum of the probabilities of all outcomes up to and including a certain event value in a probability distribution. It represents the probability that a random variable is less than or equal to a particular value. Within the steps provided from our exercise, cumulative probability played a pivotal role while calculating how likely it is to observe fewer than a certain number of births within an hour.

Using cumulative probability helps to answer questions about intervals or ranges, such as 'What is the probability that there are fewer than 3 births in a given hour?' or 'What range of hourly births will we see at least 89% of the time?' This is a powerful aspect of probability theory since it allows for decision making based on threshold values and can inform strategies in scheduling, risk management, and resource allocation.

Cumulative probability in the context of the Poisson distribution is particularly insightful for understanding the flow of events over time and can provide a clearer picture of what to expect in various scenarios, such as the operational requirements of a hospital maternity ward.