LOS H requires us to:
calculate and interpret an unconditional probability using the total probability rule.
a. If we are given the historical empirical observations about certain data, and we are asked to give a probabilistic figure of what the expected value of such data might be for the next period. We can simply do that by calculating the equally-weighted average of such data, or its mean. In this case, the expected value would be:
b. Alternatively, we can also make a forecast by assigning certain probabilistic weights to the past observation (indicating the chances of them happening in the future) and multiplying those observation values with their probabilities. In this case, the expected value of the future would be:
c. Thus we can write the equation for the expected value as follows:
d. So, we calculated the possible value of the observation X, based on the above formula. But the above formula does not give us information as to how variable; the expected value might be from its weighted average. Thus, to find out the variability of the expected value around its average, we need to calculate the variance. The variance can be calculated using the following formula:
e. For example, consider the following set of observations and their expected probabilities:
X |
P(X) |
2.60 |
0.15 |
2.45 |
0.45 |
2.20 |
0.24 |
2.00 |
0.16 |
Given this information, we can calculate the expected value X as:
Also, we can calculate the variability around such expected value around the mean of 2.3405, as follows:
And, the standard deviation of this would be the square root of the variance: