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.）简答题
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Question

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.）简答题

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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.

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