Lab 00 - Sandbox¶

Goal: refresh basic Python operations used in later labs.

This notebook visualizes the six key probability distributions covered in the Probability Distributions Guide.

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Quick note: Discrete vs Continuous¶

• Discrete distributions model countable outcomes (0, 1, 2, ...).
• Continuous distributions model values over a range (time, latency, temperature).
• In this notebook: Bernoulli, Binomial, and Poisson are Discrete; Normal and Exponential are Continuous; Uniform is shown as a Discrete example.

In [6]:

from scipy.stats import binom
from scipy.stats import norm
from scipy.stats import expon
from scipy.stats import poisson
import matplotlib.pyplot as plt
import numpy as np

from scipy.stats import binom from scipy.stats import norm from scipy.stats import expon from scipy.stats import poisson import matplotlib.pyplot as plt import numpy as np

Uniform Distribution (Discrete Example)¶

Every outcome has the same probability. At very high temperature, LLM token selection approaches this — perfectly flat, maximally random.

In [7]:

outcomes = 10
probs = np.ones(outcomes) / outcomes
plt.bar(range(outcomes), probs, color='steelblue', alpha=0.7, edgecolor='black')
plt.xlabel('Outcome')
plt.ylabel('Probability')
plt.title('Uniform Distribution (10 equally likely outcomes)')
plt.ylim(0, 0.2)
plt.grid(axis='y', alpha=0.3)
plt.show()

outcomes = 10 probs = np.ones(outcomes) / outcomes plt.bar(range(outcomes), probs, color='steelblue', alpha=0.7, edgecolor='black') plt.xlabel('Outcome') plt.ylabel('Probability') plt.title('Uniform Distribution (10 equally likely outcomes)') plt.ylim(0, 0.2) plt.grid(axis='y', alpha=0.3) plt.show()

Bernoulli Distribution (Discrete)¶

A single yes/no trial. One classification decision — was the route correct or not — is Bernoulli.

In [8]:

outcomes = ['Failure', 'Success']
probs = [0.3, 0.7]
plt.bar(outcomes, probs, color=['#ff6b6b', '#51cf66'], alpha=0.7, edgecolor='black')
plt.ylabel('Probability')
plt.title('Bernoulli Distribution (p=0.7)')
plt.ylim(0, 1)
plt.grid(axis='y', alpha=0.3)
plt.show()

outcomes = ['Failure', 'Success'] probs = [0.3, 0.7] plt.bar(outcomes, probs, color=['#ff6b6b', '#51cf66'], alpha=0.7, edgecolor='black') plt.ylabel('Probability') plt.title('Bernoulli Distribution (p=0.7)') plt.ylim(0, 1) plt.grid(axis='y', alpha=0.3) plt.show()

Binomial Distribution (Discrete)¶

Counts successes across many Bernoulli trials. How many correct routes in a batch of 20 requests?

In [9]:

n, p = 20, 0.7
x = range(0, n+1)
probs = [binom.pmf(k, n, p) for k in x]
plt.bar(x, probs, color='teal', alpha=0.7, edgecolor='black')
plt.xlabel('Number of Successes')
plt.ylabel('Probability')
plt.title('Binomial Distribution (n=20, p=0.7)')
plt.grid(axis='y', alpha=0.3)
plt.show()

n, p = 20, 0.7 x = range(0, n+1) probs = [binom.pmf(k, n, p) for k in x] plt.bar(x, probs, color='teal', alpha=0.7, edgecolor='black') plt.xlabel('Number of Successes') plt.ylabel('Probability') plt.title('Binomial Distribution (n=20, p=0.7)') plt.grid(axis='y', alpha=0.3) plt.show()

Normal (Gaussian) Distribution (Continuous)¶

Bell-curve. Aggregated metrics (average scores, latencies across many requests) tend to converge toward a normal distribution.

In [10]:

mu, sigma = 0, 1
x = np.linspace(-4, 4, 100)
y = norm.pdf(x, mu, sigma)
plt.plot(x, y, linewidth=2.5, color='darkviolet')
plt.fill_between(x, y, alpha=0.3, color='darkviolet')
plt.xlabel('Value')
plt.ylabel('Probability Density')
plt.title('Normal Distribution (μ=0, σ=1)')
plt.grid(alpha=0.3)
plt.show()

mu, sigma = 0, 1 x = np.linspace(-4, 4, 100) y = norm.pdf(x, mu, sigma) plt.plot(x, y, linewidth=2.5, color='darkviolet') plt.fill_between(x, y, alpha=0.3, color='darkviolet') plt.xlabel('Value') plt.ylabel('Probability Density') plt.title('Normal Distribution (μ=0, σ=1)') plt.grid(alpha=0.3) plt.show()

Exponential Distribution (Continuous)¶

Models waiting time between events. How long until the next escalation or incident arrives?

In [11]:

lam = 2
t = np.linspace(0, 3, 100)
y = expon.pdf(t, scale=1/lam)
plt.plot(t, y, linewidth=2.5, color='coral')
plt.fill_between(t, y, alpha=0.3, color='coral')
plt.xlabel('Time')
plt.ylabel('Probability Density')
plt.title('Exponential Distribution (λ=2)')
plt.grid(alpha=0.3)
plt.show()

lam = 2 t = np.linspace(0, 3, 100) y = expon.pdf(t, scale=1/lam) plt.plot(t, y, linewidth=2.5, color='coral') plt.fill_between(t, y, alpha=0.3, color='coral') plt.xlabel('Time') plt.ylabel('Probability Density') plt.title('Exponential Distribution (λ=2)') plt.grid(alpha=0.3) plt.show()

Poisson Distribution (Discrete)¶

Counts how many events happen in a fixed window. Incidents per hour, errors per 1,000 requests.

In [12]:

lam = 5
k = range(0, 15)
probs = [poisson.pmf(x, lam) for x in k]
plt.bar(k, probs, color='darkgreen', alpha=0.7, edgecolor='black')
plt.xlabel('Count')
plt.ylabel('Probability')
plt.title('Poisson Distribution (λ=5)')
plt.grid(axis='y', alpha=0.3)
plt.show()

lam = 5 k = range(0, 15) probs = [poisson.pmf(x, lam) for x in k] plt.bar(k, probs, color='darkgreen', alpha=0.7, edgecolor='black') plt.xlabel('Count') plt.ylabel('Probability') plt.title('Poisson Distribution (λ=5)') plt.grid(axis='y', alpha=0.3) plt.show()

Temperature Effects on Token Probability Distribution (Categorical / Discrete)¶

Same logits, four different temperatures. Watch how probability mass spreads as temperature rises — from sharp and decisive to near-uniform.

In [13]:

logits = np.array([5, 3, 1, 0.5, 0.1])

fig, axes = plt.subplots(1, 4, figsize=(14, 3))
temperatures = [0.5, 1.0, 1.5, 2.0]

for ax, T in zip(axes, temperatures):
    probs = np.exp(logits / T) / np.sum(np.exp(logits / T))
    ax.bar(range(5), probs, color='steelblue', alpha=0.7, edgecolor='black')
    ax.set_title(f'Temperature = {T}')
    ax.set_ylim(0, 1)
    ax.set_xlabel('Token index')
    ax.set_ylabel('Probability' if T == 0.5 else '')
    ax.grid(axis='y', alpha=0.3)

plt.suptitle('Effect of Temperature on Token Probability Distribution', fontsize=12, y=1.02)
plt.tight_layout()
plt.show()

logits = np.array([5, 3, 1, 0.5, 0.1]) fig, axes = plt.subplots(1, 4, figsize=(14, 3)) temperatures = [0.5, 1.0, 1.5, 2.0] for ax, T in zip(axes, temperatures): probs = np.exp(logits / T) / np.sum(np.exp(logits / T)) ax.bar(range(5), probs, color='steelblue', alpha=0.7, edgecolor='black') ax.set_title(f'Temperature = {T}') ax.set_ylim(0, 1) ax.set_xlabel('Token index') ax.set_ylabel('Probability' if T == 0.5 else '') ax.grid(axis='y', alpha=0.3) plt.suptitle('Effect of Temperature on Token Probability Distribution', fontsize=12, y=1.02) plt.tight_layout() plt.show()