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bookChallenge: Quality Control Sampling

You are the quality control manager at a rod manufacturing factory. You need to simulate measurements and defect counts using three different probability distributions to model your production process:

  • Normal distribution for rod weights (continuous);
  • Binomial distribution for the number of defective rods in batches (discrete);
  • Uniform distribution for rod length tolerances (continuous).
Note
Note

Your task is to translate the formulas and concepts from your lecture into Python code. You must NOT use built-in numpy random sampling functions (e.g., np.random.normal) or any other library's direct sampling methods for the distributions. Instead, implement sample generation manually using the underlying principles and basic Python (e.g., random.random(), random.gauss()).

Formulas to Use

Normal distribution PDF:

f(x)=1Οƒ2Ο€eβˆ’(xβˆ’ΞΌ)22Οƒ2f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^2}{2\sigma^2}}

Standard deviation from variance:

Οƒ=variance\sigma = \sqrt{\text{variance}}

Binomial distribution PMF:

P(X=k)=(nk)nk(1βˆ’n)nβˆ’k,where(nk)=n!k!(nβˆ’k)!P(X = k) = \begin{pmatrix}n\\k\end{pmatrix}n^k(1-n)^{n-k},\quad \text{where}\begin{pmatrix}n\\k\end{pmatrix} = \frac{n!}{k!(n-k)!}

Uniform distribution PDF:

f(x)=1bβˆ’afora≀x≀bf(x) = \frac{1}{b-a}\quad \text{for}\quad a \le x \le b
Task

Swipe to start coding

  1. Complete the starter code below by filling in the blanks (____) using the concepts/formulas above.
  2. Use only random and math modules.
  3. Implement three functions to generate 1000 samples from each distribution (Normal: using random.gauss(); Binomial: simulating n Bernoulli trials; Uniform: scaling random.random()).
  4. Plot histograms for each distribution (plotting code given, just complete the sampling functions and parameters).
  5. Retain all comments exactly as shown, they explain each step.
  6. No use of numpy random functions or external sampling libraries.

Solution

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SectionΒ 5. ChapterΒ 12
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bookChallenge: Quality Control Sampling

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You are the quality control manager at a rod manufacturing factory. You need to simulate measurements and defect counts using three different probability distributions to model your production process:

  • Normal distribution for rod weights (continuous);
  • Binomial distribution for the number of defective rods in batches (discrete);
  • Uniform distribution for rod length tolerances (continuous).
Note
Note

Your task is to translate the formulas and concepts from your lecture into Python code. You must NOT use built-in numpy random sampling functions (e.g., np.random.normal) or any other library's direct sampling methods for the distributions. Instead, implement sample generation manually using the underlying principles and basic Python (e.g., random.random(), random.gauss()).

Formulas to Use

Normal distribution PDF:

f(x)=1Οƒ2Ο€eβˆ’(xβˆ’ΞΌ)22Οƒ2f(x) = \frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x - \mu)^2}{2\sigma^2}}

Standard deviation from variance:

Οƒ=variance\sigma = \sqrt{\text{variance}}

Binomial distribution PMF:

P(X=k)=(nk)nk(1βˆ’n)nβˆ’k,where(nk)=n!k!(nβˆ’k)!P(X = k) = \begin{pmatrix}n\\k\end{pmatrix}n^k(1-n)^{n-k},\quad \text{where}\begin{pmatrix}n\\k\end{pmatrix} = \frac{n!}{k!(n-k)!}

Uniform distribution PDF:

f(x)=1bβˆ’afora≀x≀bf(x) = \frac{1}{b-a}\quad \text{for}\quad a \le x \le b
Task

Swipe to start coding

  1. Complete the starter code below by filling in the blanks (____) using the concepts/formulas above.
  2. Use only random and math modules.
  3. Implement three functions to generate 1000 samples from each distribution (Normal: using random.gauss(); Binomial: simulating n Bernoulli trials; Uniform: scaling random.random()).
  4. Plot histograms for each distribution (plotting code given, just complete the sampling functions and parameters).
  5. Retain all comments exactly as shown, they explain each step.
  6. No use of numpy random functions or external sampling libraries.

Solution

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SectionΒ 5. ChapterΒ 12
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