位置39的问题 The random variable x is the number of occurrences of an event over an interval of 10 minutes. It can be assumed the probability of an occurrence is the same in any two time periods of an equal length. It is known that the mean number of occurrences in 10 minutes is 5.3. The probability there are 8 occurrences in 10 minutes is 题库 1/1[input]. (Note: Round your answer to three decimal places.）

Question

BlackTom AI · Accepted Answer

The problem describes a Poisson process with a mean number of occurrences λ = 5.3 in a 10-minute interval, so the distribution of X, the number of occurrences, is Poisson(λ = 5.3).

We want P(X = 8) for a Poisson random variable, which is given by the formula P(X = k) = e^(−λ) * λ^k / k!, with k = 8 and λ = 5.3.

Compute step by step:
- Compute e^(−λ): e^(−5.3) ≈ 0.00499 (approximately; using a calculator or table).
- Compute λ^k: 5.3^8 ≈ 6.24 × 10^5.
- Compute k!: 8! = 40,320.

Now assemble: P(X = 8) ≈ 0.00499 * (6.24 × 10^5) / 40,320.
- Multiply numerator: 0.00499 * 6.24 × 10^5 ≈ 3.10 × 10^3.
- Divide by 40,320: 3.10 × 10^3 / 4.032 × 10^4 ≈ 0.0769.

Rounding to three decimal places gives P(X = 8) ≈ 0.077.

Thus, the probability of observing exactly 8 occurrences in 10 minutes is about 0.077. If any intermediate rounding were introduced earlier, it would slightly affect the final three-decimal result, but with standard calculation this is the expected value.

Note: The reasoning hinges on the Poisson assumption with a constant rate over equal intervals, which justifies using the Poisson probability mass function for X = 8.

查看解析

登录即可查看完整答案

类似问题

位置13的问题 A property of the Poisson distribution is that the mean equals the 题库 1/1[input].

In the textile industry, a manufacturer is interested in the number of blemishes or flaws occurring in each 100 feet of material. The probability distribution that has the greatest chance of applying to this situation is the

The number of visible defects on a product container is thought to be Poisson distributed with a mean equal to 3.5 defects per container. Based on this, the probability that 2 containers will contain a total of less than 2 defects is:

The number of customers who enter a bank is thought to be Poisson distributed with a mean equal to 10 per hour. What are the chances that no customers will arrive in a 15-minute period?

The number of weeds that remain living after a specific chemical has been applied averages 1.3 per square yard and follows a Poisson distribution. Based on this, what is the probability that a 3-square yard section will contain less than 4 weeds?

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单