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Chapter 12: Problem 1

Suppose that a random variable \(X\) has a Poisson distribution with \(\lambda=3,\) as in Example \(1 .\) Compute the probabilities \(P_{6}, P_{7}, P_{3}.\)

Short Answer

Expert verified
P6 鈮� 0.0504, P7 鈮� 0.0216, P3 鈮� 0.2240

Step by step solution

01

- Understand the Poisson Distribution

The Poisson distribution formula is given by \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \]where \(k\) is a non-negative integer, and \(\lambda\) is the average rate (mean) of occurrence.
02

- Insert \(\lambda\) Value

In this problem, \(\lambda = 3\). Substitute \(\lambda\) into the formula: \[ P(X = k) = \frac{e^{-3} 3^k}{k!} \]
03

- Compute \(P_6\)

To find the probability \(P_6\), set \(k = 6\): \[ P_6 = \frac{e^{-3} 3^6}{6!} \]Calculate the value: \[ P_6 = \frac{e^{-3} 729}{720} \approx 0.0504 \]
04

- Compute \(P_7\)

To find the probability \(P_7\), set \(k = 7\): \[ P_7 = \frac{e^{-3} 3^7}{7!} \]Calculate the value: \[ P_7 = \frac{e^{-3} 2187}{5040} \approx 0.0216 \]
05

- Compute \(P_3\)

To find the probability \(P_3\), set \(k = 3\): \[ P_3 = \frac{e^{-3} 3^3}{3!} \]Calculate the value: \[ P_3 = \frac{e^{-3} 27}{6} \approx 0.2240 \]

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Key Concepts

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

probability
Probability is a measure of how likely an event is to happen. It ranges between 0 and 1, where 0 means the event will not happen, and 1 means it will certainly happen. In a Poisson distribution, probability helps in determining the likelihood of a given number of events occurring within a fixed interval of time or space. Understanding probability makes it easier to use statistical formulas, such as the Poisson formula, to solve real-world problems. For example, if we want to know the chances of a random variable equalling a specific value in a Poisson distribution, we calculate its probability using the given formula.
random variable
A random variable is a variable that can take on different values, each with an associated probability. In the context of a Poisson distribution, the random variable often represents the number of events occurring within a specific interval. For instance, in the given exercise, the random variable \(X\) represents the number of events happening with an average rate of 3 (\( \lambda = 3 \)). This random variable \(X\) can take on non-negative integer values, and we use the Poisson formula to find the probability of \(X\) taking specific values like 6, 7, or 3.
mean of occurrence
The mean of occurrence, denoted by \( \lambda \) in a Poisson distribution, represents the average number of times an event happens in a fixed interval of time or space. In the example provided, the mean of occurrence is 3, implying that, on average, we expect 3 events to occur in the given interval. The mean is a crucial part of the Poisson formula and directly impacts the probabilities of different outcomes. When \( \lambda \) is known, it helps in simplifying computations and understanding the distribution of events.
Poisson formula
The Poisson formula is a statistical tool used to compute the probability of a given number of events occurring within a fixed interval. The formula is expressed as \[ P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \] where \(P(X = k)\) is the probability of \(X\) (the random variable) equaling \(k\) (the specific number of events), \(\lambda\) is the mean of occurrence, and \(e\) is Euler's number (approximately equal to 2.71828). By substituting \(\lambda = 3\) into the formula, we calculated the probabilities for specific values (e.g., \(P_6\), \(P_7\), \(P_3\)). Using the Poisson formula allows us to determine these probabilities precisely, providing valuable insights in various fields, from engineering to healthcare.

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Most popular questions from this chapter

The quality-control department at a sewing machine factory has determined that 1 out of 40 machines does not pass inspection. Let \(X\) be the number of machines on an assembly line that pass inspection before a machine is found that fails inspection. (a) Write the formula for \(\operatorname{Pr}(X=n).\) (b) What is the probability that, of five machines coming oft the assembly line, the first four pass inspection and the fifth does not?

Consider a group of patients who have been treated for an acute disease such as cancer, and let \(X\) be the number of years a person lives after receiving the treatment (the survival time). Under suitable conditions, the density function for \(X\) will be \(f(x)=k e^{-k x}\) for some constant \(k.\) (a) The survival function \(S(x)\) is the probability that a person chosen at random from the group of patients survives until at least time \(x .\) Explain why \(S(x)=1-F(x),\) where \(F(x)\) is the cumulative distribution function for \(X,\) and compute \(S(x).\) (b) Suppose that the probability is .90 that a patient will survive at least 5 years \([S(5)=.90] .\) Find the constant \(k\) in the exponential density function \(f(x).\)

Show that the function \(f(x)=e^{-x^{2} / 2}\) has a relative maximum at \(x=0.\)

In a study of the vacancies occurring in the U.S. Supreme Court, it has been determined that the time elapsed between successive resignation is an exponential random variable with expected value 2 years. Find the probability that the composition of the U.S. Supreme Court will remain unchanged for a period of 5 years or more.

The number of accidents per week at a busy intersection was recorded for a year. There were 11 weeks with no accidents, 26 weeks with one accident, 13 weeks with two accidents, and 2 weeks with three accidents. A week is to be selected at random and the number of accidents noted. Let \(X\) be the outcome. Then, \(X\) is a random variable taking on the values \(0,1,2,\) and \(3.\) (a) Write out a probability table for \(X\). (b) Compute \(E(X).\) (c) Interpret \(\mathrm{E}(X).\)
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