正态分布样本最小值的分布与期望估算技术咨询

Hey there, let's adjust your approach to get the answers you need. Generating individual bids with NORM.INV(Rand(), 10, 3.2) is a solid start, but to figure out the minimum bid's distribution and expected value, we need to focus on the smallest value from each set of 15 bids (this is called the 1st order statistic for your sample).

1. Quick Theory Refresher #

For 15 independent, identical normal variables (your bids), here's the key math to keep in mind:

• The cumulative distribution function (CDF) of the minimum bid tells you the probability that the minimum is ≤ some value x:
  F_min(x) = 1 - [1 - F(x)]^15
  Where F(x) is the CDF of your normal distribution (use Excel's NORM.DIST(x,10,3.2,TRUE) to calculate this).
• The probability density function (PDF) of the minimum (for plotting the distribution shape) is:
  f_min(x) = 15 * [1 - F(x)]^14 * f(x)
  Here, f(x) is the normal PDF (NORM.DIST(x,10,3.2,FALSE)).

The catch? There's no simple closed-form formula for the expected value of the minimum of normal variables. So we'll use simulation (the easiest, most intuitive method in Excel) or numerical integration for more precision.

2. Step-by-Step Excel Simulation #

This builds directly on what you're already doing, just scaled up to get meaningful data:

• Generate multiple bid sets: In cells A1:O1, enter =NORM.INV(RAND(),10,3.2) in each cell to create one set of 15 bids. Drag this formula down to row 10,000 (more rows = more accurate results—10k is a great balance of speed and reliability).
• Calculate the minimum for each set: In cell P1, use =MIN(A1:O1) to get the smallest bid from the first group. Drag this down to row 10,000 to collect all your minimum bids.
• Analyze the results:
  • Expected Value: Run =AVERAGE(P:P)—this gives you the mean of all simulated minimum bids. For your parameters, you should get a value around 3.7-3.9 with 10k runs.
  • Distribution Shape: Insert a histogram for column P—you'll see it's left-skewed, since the minimum can't dip far below the original distribution's lower tail but is constrained on the high end.
  • Key Percentiles: Use =PERCENTILE.EXC(P:P, 0.05) to find the 5th percentile (the bid value where only 5% of minimums are lower) or other percentiles to understand the spread of possible minimums.

3. Numerical Integration (For Tighter Precision) #

If you want a more precise estimate without running thousands of simulations, you can approximate the expected value via integral:
E[X_min] = ∫₋∞^∞ x * 15 * [1 - NORM.DIST(x,10,3.2,TRUE)]^14 * NORM.DIST(x,10,3.2,FALSE) dx
In Excel, do this with a sum over small intervals:

• Create a column of x-values from -5 to 10 in 0.1 increments.
• For each x, calculate the integrand using the formula above.
• Multiply each integrand by the interval size (0.1) and sum all those products—this gives you an approximate expected value.

内容的提问来源于stack exchange，提问作者nojohnny101