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Chapter 5: Problem 57

An average of \(.8\) accident occur per day in a particular large city. a. Find the probability that no accident will occur in this city on ? given day. b. Let \(x\) denote the number of accidents that will occur in this city on a given day. Write the probability distribution of \(x\). c. Find the mean, variance, and standard deviation of the probability distribution developed in part b.

Short Answer

Expert verified
The probability that no accident will occur in this city on a given day is calculated through the Poisson distribution formula as described in Step 1. The probability distribution of accidents on a given day is given as \( P(X=x) = \frac{e^{-0.8} \cdot 0.8^x}{x!}\) for \( x = 0, 1, 2, 3,....\). The mean and variance are both 0.8, and the standard deviation is the square root of 0.8.

Step by step solution

01

Calculation of Probability

The formula of the Poisson distribution P(X=x) is given by: \[ P(X = x) = \frac{e^{-\lambda} \cdot \lambda^x}{x!}\] Substituting the provided average value \( \lambda = 0.8 \) and the desired days \( x = 0 \) into the equation, the result is: \[ P(X = 0) = \frac{e^{-0.8} \cdot 0.8^0}{0!} \]
02

Derivation of the Probability Distribution of Accidents

The probability distribution for a given day can be expressed in the form of P(X = x) represented with the Poisson distribution formula: \[ P(X=x) = \frac{e^{-0.8} \cdot 0.8^x}{x!} \] for \( x = 0, 1, 2, 3,....\).
03

Computation of Mean, Variance, and Standard Deviation

For a Poisson distribution, the mean, variance, and standard deviation are given by \(\lambda\), \(\lambda\), and \( \sqrt{\lambda} \) respectively. So, substituting \( \lambda = 0.8 \) into these formulas, the mean, variance, and standard deviation can be calculated.

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Key Concepts

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

Probability Theory
Probability theory is a fundamental concept used to understand random events and uncertainties. It is the mathematical framework that allows us to calculate the likelihood of different outcomes in uncertain situations. A Poisson distribution, used in this exercise, models the probability of a specific number of events occurring within a fixed period of time or space.

Imagine you want to figure out how often accidents happen in a city each day. By employing probability theory with the Poisson distribution, you can calculate the likelihood of having zero accidents.
  • The formula for the Poisson distribution is: \[ P(X = x) = \frac{e^{-\lambda} \cdot \lambda^x}{x!} \]
  • Here, \( \lambda \) is the average number of events (accidents in this case) expected in a given time frame.
  • The value \( x \) represents the number of events you are interested in. If you want to know the probability of no accidents, you would set \( x = 0 \).
  • \( e \) is the base of natural logarithms, approximately equal to 2.71828.
Probability theory simplifies the process of calculating risks and making informed decisions in everyday life.
Mean and Variance
In any probability distribution, the mean and variance are crucial concepts. They help describe the central tendency and spread of the distribution. For a Poisson distribution, both the mean and variance are incredibly straightforward, as they both equal the parameter \( \lambda \), the average number of occurrences. Let's break this down.

  • Mean: The mean of a Poisson distribution indicates the expected average number of events. Here, \( \lambda = 0.8 \), so on average, 0.8 accidents are expected per day in this city. This might seem odd for a number of accidents, but remember it's a theoretical average.
  • Variance: The variance for a Poisson distribution is also equal to \( \lambda \). It measures how much the number of accidents will vary around the mean. In this case, the variance is 0.8, suggesting that events are moderately spread out around the mean.
  • The Poisson distribution's simplicity, with equal mean and variance, makes it easy to work with, especially when predicting event frequencies over time.
Standard Deviation
The standard deviation is another important statistical measure that gives insight into the data distribution. It calculates the amount of variation or dispersion from the mean. In a Poisson distribution, the standard deviation is the square root of the variance. Here鈥檚 what you need to know:

  • Given the variance \( \lambda = 0.8 \), the standard deviation is \( \sqrt{0.8} \).
  • This calculation results in approximately 0.894, providing a sense of the average deviation from the mean of the distribution.
  • While the variance informs us about how numbers fluctuate within the distribution, the standard deviation gives a more human-readable scale of that fluctuation. It's one of the key statistics used to determine the consistency or variability of the data.
Understanding the standard deviation, along with the mean and variance, paints a complete picture of your data, helping you predict and manage uncertainty effectively.

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