Python中泊松/正态与Beta分布实现及统计结果解读求助

Hey there! Let's walk through this together—you're working with vehicle arrival data (63 cars per minute on average, 25 std dev) using Poisson, Normal, and Beta distributions in Python, and you've got questions about the outputs and plots. Let's break it down step by step, starting with filling in the obvious gaps in your code.

First: Let's formalize your base code #

Based on your scenario, here's the core variable setup you likely have (or should have):

import scipy as sp
import numpy as np
import matplotlib.pyplot as plt

# Your given parameters: average cars per minute, standard deviation
mu, sigma = 63, 25

Now let's unpack each distribution—what it's doing, what its outputs mean, and why you might be confused by the plots.

1. Poisson Distribution: For counting independent events #

What it's for #

Poisson is built for scenarios where you count how many times a random, independent event happens in a fixed window (like cars arriving each minute). The rule here is: events don't affect each other, and the average rate stays consistent.

Example code to generate & plot #

# Generate 10,000 simulated arrival counts
poisson_samples = np.random.poisson(lam=mu, size=10000)

# Plot the sample histogram + theoretical Poisson curve
plt.figure(figsize=(10,6))
# Histogram shows how often each count occurred (normalized to probabilities)
count, bins, ignored = plt.hist(poisson_samples, bins=30, density=True, alpha=0.6, label='Simulated Arrivals')
# Calculate the theoretical probability mass function (PMF)
x = np.arange(sp.stats.poisson.ppf(0.01, mu), sp.stats.poisson.ppf(0.99, mu))
pmf = sp.stats.poisson.pmf(x, mu)
plt.plot(x, pmf, 'r-', lw=2, label='Poisson Theoretical Curve')
plt.title('Poisson Distribution: Vehicle Arrivals (μ=63)')
plt.xlabel('Number of Cars per Minute')
plt.ylabel('Probability')
plt.legend()
plt.show()

What your output/plot is telling you #

• The histogram is a snapshot of 10k simulated minutes—how often we saw 50 cars, 60 cars, etc.
• The red curve is what math predicts the distribution should look like.
• The big gotcha: Poisson's standard deviation is always the square root of the mean. For μ=63, that's ~7.94—but your real data has a std dev of 25. That means your actual arrivals are way more variable than Poisson expects! If your plot looks way tighter than your real data, that's why—your arrivals probably aren't fully independent (think rush hour clusters, traffic jams causing lulls, etc.). Poisson might not be the best fit here.

2. Normal Distribution: Approximating symmetric data #

What it's for #

Normal (Gaussian) is the classic "bell curve" for continuous, symmetric data. Since car counts are discrete integers, we're using it as an approximation here. It works well if your data is roughly symmetric and has that familiar hump shape.

Example code to generate & plot #

# Generate normal samples, round to integers (since you can't have half a car!)
normal_samples = np.random.normal(loc=mu, scale=sigma, size=10000)
normal_samples = np.round(normal_samples).astype(int)
# Filter out negative numbers—you can't have negative cars arriving!
normal_samples = normal_samples[normal_samples >= 0]

# Plot the histogram + theoretical normal curve
plt.figure(figsize=(10,6))
count, bins, ignored = plt.hist(normal_samples, bins=30, density=True, alpha=0.6, label='Simulated Arrivals')
# Calculate the theoretical probability density function (PDF)
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
pdf = sp.stats.norm.pdf(x, mu, sigma)
plt.plot(x, pdf, 'g-', lw=2, label='Normal Theoretical Curve')
plt.title('Normal Distribution: Vehicle Arrivals (μ=63, σ=25)')
plt.xlabel('Number of Cars per Minute')
plt.ylabel('Probability Density')
plt.legend()
plt.show()

What your output/plot is telling you #

• The histogram shows your simulated counts, with weird negative values stripped out.
• The green curve is the ideal bell shape centered at 63, spanning from ~63-75 to ~63+75 (the 3σ range, where most data lives).
• Common confusion points:
  1. If you see negative values in your raw samples—don't panic! Normal distribution allows negatives, but we just filter them out because they don't make sense for car counts.
  2. If your real data has more low-count minutes than the bell curve predicts—normal might not capture the skew in your actual arrivals. For that, you might want to look into a truncated normal distribution (scipy.stats.truncnorm) instead.

3. Beta Distribution: For proportions, not raw counts #

What it's for #

Beta is for data that's a proportion between 0 and 1—like the percentage of time a lane is occupied, or the ratio of cars to max capacity. It doesn't make sense for raw car counts (which can be way over 1) unless you normalize your data first.

Example code (with proper normalization) #

# First, pick a realistic max number of cars per minute (adjust this to your actual data!)
max_vehicles = 200
# Convert your mean to a proportion (63/200 = 0.315)
mu_norm = mu / max_vehicles
# Convert your std dev to a proportion too
var_norm = (sigma ** 2) / (max_vehicles ** 2)

# Calculate Beta's shape parameters (α and β) using the normalized mean/variance
alpha = mu_norm * ((mu_norm*(1 - mu_norm))/var_norm - 1)
beta = (1 - mu_norm) * ((mu_norm*(1 - mu_norm))/var_norm - 1)

# Generate Beta samples, then convert back to raw car counts
beta_samples = np.random.beta(alpha, beta, size=10000)
beta_samples = np.round(beta_samples * max_vehicles).astype(int)

# Plot the histogram + scaled Beta curve
plt.figure(figsize=(10,6))
count, bins, ignored = plt.hist(beta_samples, bins=30, density=True, alpha=0.6, label='Simulated Arrivals')
# Calculate theoretical PDF (scaled back to car counts)
x_norm = np.linspace(0, 1, 100)
pdf_norm = sp.stats.beta.pdf(x_norm, alpha, beta)
# Scale the PDF to match raw car counts
x = x_norm * max_vehicles
pdf = pdf_norm / max_vehicles
plt.plot(x, pdf, 'b-', lw=2, label='Beta Theoretical Curve (Scaled)')
plt.title('Beta Distribution: Vehicle Arrivals (Normalized to 0-200 Cars)')
plt.xlabel('Number of Cars per Minute')
plt.ylabel('Probability Density')
plt.legend()
plt.show()

What your output/plot is telling you #

• The histogram shows counts converted back from the 0-1 proportion range.
• The blue curve is the Beta distribution scaled to fit your car count range.
• The big confusion here: If your Beta plot looks weird, it's almost certainly because you didn't normalize your data first. Beta can't handle raw counts directly—you have to anchor it to a max possible value. Pick a max that makes sense for your city (maybe peak hour max) and adjust accordingly.

Quick recap of common questions you might have #

• Why is my Poisson plot way tighter than my real data? Because Poisson's std dev is fixed at √μ, and your real data is way more variable. Poisson isn't a good fit here.
• Why does my Normal distribution have negative numbers? It's a continuous distribution—just filter those out, or use a truncated normal instead.
• Why does my Beta distribution look nothing like my data? You forgot to normalize your counts to a 0-1 proportion first. Pick a realistic max car count and try again.

内容的提问来源于stack exchange，提问作者Giroud2