1. The problem is to generate three new random values based on the given values $0.8551459$, $0.9577651$, and $1.07$ such that the new values do not exceed the original upper and lower limits and their average remains the same.
2. First, calculate the average of the original values:
$$\text{average} = \frac{0.8551459 + 0.9577651 + 1.07}{3} = \frac{2.882911}{3} = 0.9609703333$$
3. The lower limit is the minimum of the original values: $0.8551459$.
4. The upper limit is the maximum of the original values: $1.07$.
5. To generate new values $x$, $y$, and $z$ such that:
- $0.8551459 \leq x,y,z \leq 1.07$
- $\frac{x + y + z}{3} = 0.9609703333$
6. One approach is to randomly select $x$ and $y$ within the limits, then solve for $z$:
$$z = 3 \times 0.9609703333 - x - y$$
7. Ensure $z$ also lies within the limits. If not, repeat the random selection.
8. Example:
- Let $x = 0.9$
- Let $y = 1.0$
Then:
$$z = 3 \times 0.9609703333 - 0.9 - 1.0 = 2.882911 - 1.9 = 0.982911$$
9. Check if $z$ is within limits:
$$0.8551459 \leq 0.982911 \leq 1.07$$ which is true.
10. Therefore, new values can be:
$$x = 0.9, y = 1.0, z = 0.982911$$
11. Verify the average:
$$\frac{0.9 + 1.0 + 0.982911}{3} = \frac{2.882911}{3} = 0.9609703333$$ which matches the original average.
This method ensures the new values respect the original limits and maintain the same average.
Random Values
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