What is the method for determining the mean and variance of two intervals?

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To determine the mean and variance of two intervals, follow these steps:

1. Find the midpoint of each interval by adding the lower and upper bounds of each interval and dividing by two.
2. Calculate the weighted mean of the midpoints by multiplying each midpoint by its respective interval length, adding the products, and dividing by the total length of the two intervals.
3. Calculate the weighted variance of the midpoints by adding the product of each interval length squared and its respective midpoint variance, dividing by the total length squared, and subtracting the squared weighted mean.

For example, consider the intervals [0, 5) and [10, 15). The midpoints are 2.5 and 12.5, respectively. The interval lengths are both 5, so the total length is 10. The weighted mean is (2.5 * 5 + 12.5 * 5) / 10 = 7.5. The midpoint variances are [(5 - 0)^2 / 12] and [(15 - 10)^2 / 12], respectively. The interval length squares are both 25, so the total length squared is 50. The weighted variance is [(25 * 1/12 * ((5-0)^2/12)) + (25 * 1/12 * ((15-10)^2/12))] / 50 - 7.5^2 = 8.75. Therefore, the mean and variance of the two intervals are 7.5 and 8.75, respectively.