Question

Let x have a Poisson distribution with a mean of _______.

1) p(3 ≤ x ≤ 5)
2) p(x ≥ 3)
3) p(x ≤ 3)

Answer 1

The question involves calculating various probabilities for a Poisson distributed random variable with a given mean, using Poisson probability density and cumulative distribution functions.

Step-by-step explanation:

The question pertains to the Poisson distribution, which is a probability distribution that measures the probability of a number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. In this case, the student needs to calculate probabilities for various conditions given different means for the Poisson distribution. These calculations involve using Poisson probability functions, such as poissonpdf (Poisson probability density function) and poissoncdf (Poisson cumulative distribution function).

Examples:

• To find P(x = 3) where X~Poisson(λ), we use the formula: λ^x * e^{-λ} / x!.
• The probability P(x ≥ k) can be calculated as 1 - P(x ≤ k - 1), which requires cumulative Poisson probabilities.
• The sum of probabilities for all possible occurrences in a Poisson distribution always equals 1, as it represents the total probability space.

To solve the problems posed by the student, one needs to apply these concepts correctly using either a calculator with statistical functions, a statistical software, or tables containing Poisson probabilities.