Review & Cheat Sheet
[!IMPORTANT] Key Goal: Ensure you can distinguish between Discrete and Continuous distributions and know when to apply Gaussian, Beta, or Gamma.
This review chapter consolidates the core concepts of probability distributions, including discrete variables, the Gaussian bell curve, and the continuous Beta and Gamma distributions.
1. Key Takeaways
- Discrete vs Continuous:
- Discrete (PMF): Sums to 1. Used for countable outcomes (Coin flips, Emails).
- Continuous (PDF): Integrates to 1. Area under curve = Probability. Used for measurable outcomes (Height, Time).
- The Gaussian King:
- Symmetric, Bell-shaped. Defined by Mean (μ) and Std Dev (σ).
- 68-95-99.7 Rule describes the spread.
- Z-score: Standardizes any normal distribution to N(0, 1).
- The Flexible Friends:
- Beta: Bounded [0, 1]. Models probabilities/proportions. “Conjugate Prior” for Binomial.
- Gamma: Bounded [0, ∞). Models waiting times for k events.
Quick Revision
- Review the PMF for discrete and PDF for continuous random variables.
- Use log-probabilities to avoid integer overflow and underflow.
- The Gaussian distribution naturally arises from the Central Limit Theorem.
- The Beta distribution acts as the conjugate prior for probabilities.
- The Gamma distribution effectively models waiting times for multiple events.
2. Interactive Flashcards
Test your knowledge! Click a card to flip it.
PMF vs PDF
Click to flip
PMF is for Discrete (P(X=x)). PDF is for Continuous (Area under curve).
What is a Z-Score?
Click to flip
It measures how many standard deviations a data point is from the mean. Z = (x - μ) / σ
Beta Distribution Use Case?
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Modeling probabilities or proportions (bounded [0, 1]). E.g., CTR, Conversion Rate, Bayesian Priors.
Gamma vs Exponential?
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Exponential is waiting time for 1 event. Gamma is waiting time for k events.
Why use Log-Probabilities?
Click to flip
To avoid underflow (tiny numbers becoming 0) and overflow (factorials) when calculating with many probabilities.
3. Cheat Sheet
| Distribution | Type | Range | Parameters | Interpretation |
|---|---|---|---|---|
| Bernoulli | Discrete | {0, 1} | p | Single trial (Success/Fail) |
| Binomial | Discrete | {0..n} | n, p | Count successes in n trials |
| Poisson | Discrete | {0..∞} | λ | Count events in fixed interval |
| Gaussian | Continuous | (-∞, ∞) | μ, σ | Natural variation, Sum of errors (CLT) |
| Beta | Continuous | [0, 1] | α, β | Probability of a probability |
| Gamma | Continuous | [0, ∞) | k, θ | Waiting time for k events |
4. Next Steps
Now that you understand the fundamental distributions, you’re ready to explore how they interact in higher dimensions.