Final answer:
The question involves calculating probabilities for different ranges of the random variable X using the Poisson distribution with a mean (μ) of 4. For part (a), we compute P(X≤4) and P(X<4), which involves summing Poisson probabilities. Parts (b) and (c) involve calculating probabilities over different ranges, and part (d) asks for the probability related to the mean and standard deviation.
Step-by-step explanation:
The problem is based on the Poisson distribution, which is used to model the number of events occurring within a fixed interval of time or space when these events occur with a known average rate and independently of the time since the last event. The mean (μ) is given as 4 for this problem.
To find P(X≤4) and P(X<4), we use the Poisson probability mass function:
P(X=x) = (μ^x * e^-μ) / x!
For (a), we sum the probabilities from X=0 to X=4 for P(X≤4), and we sum the probabilities from X=0 to X=3 for P(X<4). Using a calculator or software:
P(X≤4) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) ≈ 0.6288
P(X<4) is the same as P(X≤4) - P(X=4).
For (b), P(4≤X≤7) is the sum of the probabilities from X=4 to X=7.
Selecting the correct Poisson probabilities, we find:
- P(4≤X≤7) = P(X=4) + P(X=5) + P(X=6) + P(X=7)
For (c), P(7≤X) is the complement to the probability that X is less than 7, so 1 - P(X<7).
For (d), the standard deviation (σ) of a Poisson distribution is the square root of the mean, so σ = √μ. The problem asks for the probability that X does not exceed the mean by more than one standard deviation, which is the same as finding P(μ - σ ≤ X ≤ μ + σ).