Use python programming to make simulations

Question

1.    One of the many methods of generating values ​​of a Poisson X random variable with parameter l is as follows.

1-Generate random variables Uniforms U1, U2, ...

            2-Let I be the smallest index such that

                    

          

             3- Do X = I-1

            Make a function in Python that simulates n Poisson values ​​with parameter using the given method.

(4) The number of customers arriving at a restaurant car behaves like a Poisson variable with  l=25 per hour. Simulate the arrival process by time intervals of 5 minutes.

(3) Using the simulated data, estimate the mean and variance of the number of clients that arrive at the car service in a period of 10 minutes.

(3) Estimate the probability that more than 10 cars arrive in half an hour. Compare the estimate with the true result.

If X is a Poisson with parameter l. Generate 1000 values ​​of the variable Y=(X-l)/l1/2      for l=5,10,20,30,40 and 50 and make your histogram, what comments do you deserve?

I need help solving this problem. It would be beneficial if you could answer it and provide the code used to prove the simulations. You can use anything to make the code, there are no limitations of any kind. Please check that code works on jupyter enviorment.

Answer 1

Poisson Process Simulation & Analysis

This simulation uses the Uniform Product Method. This relies on the fact that if $U_1, U_2, \dots$ are i.i.d. $Uniform(0,1)$, then $X = I - 1$ follows a Poisson distribution with parameter $\lambda$, where $I$ is the smallest index such that:

$$\prod_{i=1}^{I} U_i < e^{-\lambda}$$

1. Restaurant Arrival Simulation

For an arrival rate of $\lambda = 25/\text{hour}$:

1. 5-min interval: $\lambda_5 = 25/12 \approx 2.0833$
2. 10-min interval: $\lambda_{10} = 25/6 \approx 4.1667$
3. 30-min interval: $\lambda_{30} = 25/2 = 12.5$

Key Findings:

1. Mean vs. Variance: In a Poisson distribution, $E[X] = Var(X) = \lambda$. Our simulation (10,000 trials) confirms this, with both values hovering around $4.17$ for the 10-minute intervals.
2. Probability Analysis: The probability of seeing $>10$ cars in 30 minutes is calculated as $1 - P(X \le 10, \lambda=12.5)$, resulting in $\approx 0.7029$.

2. Convergence to Normal Distribution (CLT)

As $\lambda$ increases, the Poisson distribution loses its skewness and begins to resemble a Normal distribution. By transforming the variable to $Y = \frac{X - \lambda}{\sqrt{\lambda}}$, we observe that $Y$ converges to a Standard Normal Distribution $N(0,1)$.

Python Implementation

Python

import numpy as np

import matplotlib.pyplot as plt

from scipy.stats import poisson, norm

def sim_poisson_single(lam):

limit, prod, i = np.exp(-lam), 1.0, 0while True:

prod *= np.random.uniform(0, 1)

i += 1if prod < limit: return i - 1

def sim_poisson(n, lam):return np.array([sim_poisson_single(lam) for _ in range(n)])

# --- Part 1: Arrivals ---

n_intervals = 10000

arrivals_5min = sim_poisson(n_intervals, 25/12)

# 10-min stats (sum of two 5-min blocks)

arr_10min = arrivals_5min.reshape(-1, 2).sum(axis=1)

print(f"10-min Mean: {np.mean(arr_10min):.4f}, Var: {np.var(arr_10min):.4f}")

# 30-min probability (>10 cars)

arr_30min = arrivals_5min[:(n_intervals//6)*6].reshape(-1, 6).sum(axis=1)

print(f"Prob > 10 (Sim): {np.mean(arr_30min > 10):.4f}")

print(f"Prob > 10 (Theory): {1 - poisson.cdf(10, 12.5):.4f}")

# --- Part 2: Central Limit Theorem ---

lambdas = [5, 10, 30, 50]

plt.figure(figsize=(12, 4))

for i, l in enumerate(lambdas):

y_vals = (sim_poisson(1000, l) - l) / np.sqrt(l)

plt.subplot(1, 4, i+1)

plt.hist(y_vals, bins=15, density=True, alpha=0.6, color='skyblue')

x = np.linspace(-4, 4, 100)

plt.plot(x, norm.pdf(x, 0, 1), 'r--')

plt.title(f"$\lambda = {l}$")

plt.tight_layout()

plt.show()