In this lesson, we're exploring how three common probability distributions — Binomial, Uniform, and Normal — can be applied to quality control in manufacturing.
Wel'l walk through Python code that calculates probabilities and creates visualizations for each distribution. The goal is to understand both the statistical concepts and how the code works step-by-step.

Binomial Distribution

The Binomial distribution models the probability of getting exactly $k$ successes in $n$ independent trials, each with probability $p$ of success.


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from scipy.stats import binom
import matplotlib.pyplot as plt

# number of trials
n = 100
# probability of success
p = 0.02
# number of successes
k = 3

binom_prob = binom.pmf(k, n, p)
print("Binomial probability:", binom_prob)

# Vizualization
x_vals = range(0, 15)
y_vals = binom.pmf(x_vals, n, p)
plt.bar(x_vals, y_vals, color='skyblue')
plt.show()

n = 100 - we're testing 100 rods;
p = 0.02 - 2% chance a rod is defective;
k = 3 - probability of exactly 3 defectives;
binom.pmf() computes the probability mass function.

Uniform Distribution

The Uniform distribution models a continuous variable where all values between $a$ and $b$ are equally likely.


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from scipy.stats import uniform
import numpy as np

a = 49.5
b = 50.5
low, high = 49.8, 50.2

uniform_prob = uniform.cdf(high, a, b - a) - uniform.cdf(low, a, b - a)
print("Uniform probability:", uniform_prob)

# Vizualization
x = np.linspace(a, b, 100)
pdf = uniform.pdf(x, a, b - a)
plt.plot(x, pdf, color='black')
plt.fill_between(x, pdf, where=(x >= low) & (x <= high), color='lightgreen', alpha=0.5)
plt.show()

a, b - total range of rod lengths;
low, high - interval of interest;
Subtracting CDF values gives probability inside interval.

Normal Distribution

The Normal distribution describes values clustering around a mean $\mu$ with spread measured by standard deviation $\sigma$.


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from scipy.stats import norm

mu = 200
sigma = 5
lower, upper = 195, 205

norm_prob = norm.cdf(upper, mu, sigma) - norm.cdf(lower, mu, sigma)
print("Normal probability:", norm_prob)

z1 = (lower - mu) / sigma
z2 = (upper - mu) / sigma
print("Z-scores:", z1, z2)

# Vizualization
x = np.linspace(mu - 4*sigma, mu + 4*sigma, 200)
pdf = norm.pdf(x, mu, sigma)
plt.plot(x, pdf, color='black')
plt.fill_between(x, pdf, where=(x >= lower) & (x <= upper), color='plum', alpha=0.5)
plt.show()

mu - mean rod weight;
sigma - standard deviation;
Probability - CDF difference;
Z-scores show how far bounds are from mean.

Real-World Application

Binomial - how likely is a certain number of defective rods?
Uniform - are rod lengths within tolerance?
Normal - are rod weights within expected variability?

By combining these, quality control targets defects, ensures precision, and maintains product consistency.

Quiz

**1.

**2.

**3.

4. If uniform range widened from (49.5,50.5) to (49,51), what happens to P(49.8≤X≤50.2)? A) Increases B) Decreases ✅ C) Stays the same D) Becomes 1

5. In the normal code, why compute Z-scores first? A) Normalize values for comparison with standard normal ✅ B) Make plot symmetric C) Determine skewness D) Adjust mean

6. If σ reduced from 5 to 2 in normal example, what happens to P(195≤X≤205)? A) Decreases B) Increases ✅ C) Stays same D) Doubles

7. In binomial visualization, if p increases from 0.02 to 0.10 (n=100)? A) Peak shifts right ✅ B) Peak stays at k=3 but taller C) Flat distribution D) Equal bars

**8.

9. To model defective rods and rod weights together, which pair fits best? A) Binomial and Normal ✅ B) Uniform and Normal C) Binomial and Uniform D) Normal and Uniform

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Abschnitt 5. Kapitel 11

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Binomial Distribution

The Binomial distribution models the probability of getting exactly $k$ successes in $n$ independent trials, each with probability $p$ of success.


              123456789101112131415161718
            
from scipy.stats import binom
import matplotlib.pyplot as plt

# number of trials
n = 100
# probability of success
p = 0.02
# number of successes
k = 3

binom_prob = binom.pmf(k, n, p)
print("Binomial probability:", binom_prob)

# Vizualization
x_vals = range(0, 15)
y_vals = binom.pmf(x_vals, n, p)
plt.bar(x_vals, y_vals, color='skyblue')
plt.show()

n = 100 - we're testing 100 rods;
p = 0.02 - 2% chance a rod is defective;
k = 3 - probability of exactly 3 defectives;
binom.pmf() computes the probability mass function.

Uniform Distribution

The Uniform distribution models a continuous variable where all values between $a$ and $b$ are equally likely.


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from scipy.stats import uniform
import numpy as np

a = 49.5
b = 50.5
low, high = 49.8, 50.2

uniform_prob = uniform.cdf(high, a, b - a) - uniform.cdf(low, a, b - a)
print("Uniform probability:", uniform_prob)

# Vizualization
x = np.linspace(a, b, 100)
pdf = uniform.pdf(x, a, b - a)
plt.plot(x, pdf, color='black')
plt.fill_between(x, pdf, where=(x >= low) & (x <= high), color='lightgreen', alpha=0.5)
plt.show()

a, b - total range of rod lengths;
low, high - interval of interest;
Subtracting CDF values gives probability inside interval.

Normal Distribution

The Normal distribution describes values clustering around a mean $\mu$ with spread measured by standard deviation $\sigma$.


              12345678910111213141516171819
            
from scipy.stats import norm

mu = 200
sigma = 5
lower, upper = 195, 205

norm_prob = norm.cdf(upper, mu, sigma) - norm.cdf(lower, mu, sigma)
print("Normal probability:", norm_prob)

z1 = (lower - mu) / sigma
z2 = (upper - mu) / sigma
print("Z-scores:", z1, z2)

# Vizualization
x = np.linspace(mu - 4*sigma, mu + 4*sigma, 200)
pdf = norm.pdf(x, mu, sigma)
plt.plot(x, pdf, color='black')
plt.fill_between(x, pdf, where=(x >= lower) & (x <= upper), color='plum', alpha=0.5)
plt.show()

mu - mean rod weight;
sigma - standard deviation;
Probability - CDF difference;
Z-scores show how far bounds are from mean.

Real-World Application

Binomial - how likely is a certain number of defective rods?
Uniform - are rod lengths within tolerance?
Normal - are rod weights within expected variability?

By combining these, quality control targets defects, ensures precision, and maintains product consistency.

Quiz

**1.

**2.

**3.

4. If uniform range widened from (49.5,50.5) to (49,51), what happens to P(49.8≤X≤50.2)? A) Increases B) Decreases ✅ C) Stays the same D) Becomes 1

5. In the normal code, why compute Z-scores first? A) Normalize values for comparison with standard normal ✅ B) Make plot symmetric C) Determine skewness D) Adjust mean

6. If σ reduced from 5 to 2 in normal example, what happens to P(195≤X≤205)? A) Decreases B) Increases ✅ C) Stays same D) Doubles

7. In binomial visualization, if p increases from 0.02 to 0.10 (n=100)? A) Peak shifts right ✅ B) Peak stays at k=3 but taller C) Flat distribution D) Equal bars

**8.

9. To model defective rods and rod weights together, which pair fits best? A) Binomial and Normal ✅ B) Uniform and Normal C) Binomial and Uniform D) Normal and Uniform

War alles klar?

Danke für Ihr Feedback!

Abschnitt 5. Kapitel 11

Implementing Probability Distributions to Python

Binomial Distribution

Uniform Distribution

Normal Distribution

Real-World Application

Quiz

**1.

1. In the binomial code cell, what does `n = 100` represent?

2. Which function calculates the probability of exactly k defective rods?

3. In the uniform code, what does

Awesome!

Implementing Probability Distributions to Python

Binomial Distribution

Uniform Distribution

Normal Distribution

Real-World Application

Quiz

**1.

1. In the binomial code cell, what does `n = 100` represent?

2. Which function calculates the probability of exactly k defective rods?

3. In the uniform code, what does

Implementing Probability Distributions to Python

Binomial Distribution

Uniform Distribution

Normal Distribution

Real-World Application

Quiz

**1.

1. In the binomial code cell, what does n = 100 represent?

2. Which function calculates the probability of exactly k defective rods?

3. In the uniform code, what does

Awesome!

Implementing Probability Distributions to Python

Binomial Distribution

Uniform Distribution

Normal Distribution

Real-World Application

Quiz

**1.

1. In the binomial code cell, what does n = 100 represent?

2. Which function calculates the probability of exactly k defective rods?

3. In the uniform code, what does

1. In the binomial code cell, what does `n = 100` represent?

1. In the binomial code cell, what does `n = 100` represent?