Implementing Probability Distributions to Python
Binomial Distribution
The Binomial distribution models the probability of getting exactly k successes in n independent trials, each with probability p of success.
123456789101112131415161718from 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) # Vizualization x_vals = range(0, 15) y_vals = binom.pmf(x_vals, n, p) plt.bar(x_vals, y_vals, color='skyblue') plt.title(f'Binomial probability: {binom_prob:.4f}') 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.
1234567891011121314151617from scipy.stats import uniform import matplotlib.pyplot as plt 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) # 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.title(f'Uniform probability: {uniform_prob:.1f}') 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 $\sigma$.
1234567891011121314151617181920import numpy as np import matplotlib.pyplot as plt 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) z1 = (lower - mu) / sigma z2 = (upper - mu) / sigma # 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.title(f'Normal probability: {norm_prob:.4f}\nZ-scores: {z1}, {z2}') 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.
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SectionΒ 5. ChapterΒ 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.
123456789101112131415161718from 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) # Vizualization x_vals = range(0, 15) y_vals = binom.pmf(x_vals, n, p) plt.bar(x_vals, y_vals, color='skyblue') plt.title(f'Binomial probability: {binom_prob:.4f}') 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.
1234567891011121314151617from scipy.stats import uniform import matplotlib.pyplot as plt 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) # 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.title(f'Uniform probability: {uniform_prob:.1f}') 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 $\sigma$.
1234567891011121314151617181920import numpy as np import matplotlib.pyplot as plt 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) z1 = (lower - mu) / sigma z2 = (upper - mu) / sigma # 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.title(f'Normal probability: {norm_prob:.4f}\nZ-scores: {z1}, {z2}') 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.
Everything was clear?
Thanks for your feedback!
SectionΒ 5. ChapterΒ 11
