Final answer:
The Poisson probabilities for a distribution with mean 4 are calculated for various criteria, with the standard deviation being the square root of the mean. The 68% range lies within one standard deviation from the mean, and the usual range lies within two standard deviations.
Step-by-step explanation:
For a Poisson distribution with a mean (μ) of 4, we can calculate the probabilities using the given mean and the formula for Poisson probabilities:
(a) P(x=0) = (e-4 * 40) / 0! ≈ 0.0183
(b) P(x≤2) involves adding the probabilities of x=0, x=1, and x=2.
P(x=0) + P(x=1) + P(x=2) = 0.0183 + (e-4 * 41 / 1!) + (e-4 * 42 / 2!) ≈ 0.2381
(c) P(x≥4) = 1 - P(x<4) = 1 - P(x≤4) = 1 - 0.6288 = 0.3712
(d) P(x=2 or x=3) is the sum of the probabilities of x=2 and x=3.
P(x=2) + P(x=3) = (e-4 * 42 / 2!) + (e-4 * 43 / 3!) ≈ 0.2381 + 0.1954 = 0.4335
(e) The standard deviation (σ) for a Poisson distribution is the square root of the mean, σ = √μ = √4 ≈ 2.
(f) The 68% range for a Poisson distribution is typically between (μ - σ) and (μ + σ), so between 4 - 2 and 4 + 2, which is 2 to 6.
(g) The usual range for a Poisson distribution is often considered to be from (μ - 2σ) to (μ + 2σ), hence from 0 to 8.