Final answer:
Forecast distributions provide a nuanced view of potential outcomes beyond the mean, indicating the spread and shape of future values. Multiple forecast intervals, when combined, adequately summarize the full distribution of future values, contrary to the incorrect assertions in some of the presented statements.
Step-by-step explanation:
Forecast distributions provide valuable information when analyzing the potential outcomes of a random variable. They help to understand not just the central tendency, like the mean, but the entire range of possible future values through measures of spread and the shape of the distribution.
Statement A suggests that a forecast of the mean provides more information than a forecast of the distribution, which is not true. The mean is only a measure of central tendency and doesn't capture the range or likelihood of various outcomes. Statement B reverses this and correctly states that a forecast of the distribution provides more information than just the mean, as it indicates the variability and possibilities of different outcomes.
Statement C is true because multiple forecast intervals can indeed summarize the full distribution by showing different probabilities for where the variable could lie. For example, separately stating the 50%, 90%, and 99% forecast intervals can illustrate a clear picture of the variable's potential distribution. On the other hand, Statement D is not true because it incorrectly suggests that multiple forecast intervals cannot summarize the full distribution of future values. When combined, they give a comprehensive view of the expected range and likelihood of outcomes.
Finally, it's crucial to understand the role of the Central Limit Theorem which states that the distribution of the sum (or average) of a large number of independent, identically distributed variables tends toward a normal distribution, regardless of the shape of the original distribution, given that the sample size is sufficiently large.