Final answer:
The probabilities associated with the Poisson distribution where the mean number of emails is 6 can be calculated using the Poisson probability mass function and cumulative distribution functions, with final answers rounded to four decimal places.
Step-by-step explanation:
The random variable x represents the number of emails a student receives on a day. Given that it follows a Poisson distribution with a mean of 6 emails (X ~ P(6)), we can calculate the various probabilities requested:
- A) The probability that x is exactly 2: P(x = 2) is calculated using the Poisson probability mass function (PMF). For a Poisson distribution with a mean (λ) of 6, PMF is given by P(x; λ) = (e-λ * λx) / x!, where e is the base of the natural logarithm. Thus P(x = 2; 6) = (e-6 * 62)/2!.
- B) The probability that x is at least 2: P(x ≥ 2) = 1 - P(x < 2) which is 1 minus the cumulative distribution for x = 0 and x = 1.
- C) The probability that x is less than 2: P(x < 2) = P(x = 0) + P(x = 1).
- D) The probability that x is at most 2: P(x ≤ 2) = P(x < 2) + P(x = 2) since it includes 2.
- E) The probability that x is more than 2: P(x > 2) = 1 - P(x ≤ 2).
For A, B, C, and D, one would typically use the Poisson PMF and cumulative distribution functions, possibly through a calculator or software that has these statistical functions built-in. Rounding to four decimal places as specified would provide the final probabilities.