CSE30301: Basic Math for AI – Quiz‑Ready Summary

---

1. Recap: Probability & Statistics

• Sample Mean

  • ( \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i )
  • Unbiased: (E[\bar{X}]=\mu)
  • As (n\to\infty), (\bar{X}\to\mu) (Law of Large Numbers)

• Sample Variance

  • (S^2=\frac{1}{n-1}\sum_{i=1}^{n}(X_i-\bar{X})^2)
  • Unbiased: (E[S^2]=\sigma^2)

---

2. Central Limit Theorem (CLT)

• For i.i.d. (X_i) with (E[X_i]=\mu,;Var[X_i]=\sigma^2<\infty):
  [ \sqrt{n}\left(\bar{X}-\mu\right)\xrightarrow{d}N(0,\sigma^2) ]
• Holds regardless of original distribution (illustrated with Bernoulli, Uniform, and various distributions).

---

3. Algorithm Comparison (Illustrative Example)

• Two algorithms A and B produce random variables (X) and (Y).
• Goal: determine if (E[X] < E[Y]) (or vice‑versa).
• Approach:
  1. Sampling → draw (X_1,\dots,X_n) and (Y_1,\dots,Y_n).
  2. Compute sample means (\bar{X}) and (\bar{Y}).
  3. Use statistical inference (hypothesis test or confidence interval).

---

4. Statistical Testing

4.1 Hypotheses

• Null (H_0): No difference (e.g., (E[X]=E[Y])).
• Alternative (H_1): Difference exists (e.g., (E[X]<E[Y])).

4.2 Test Statistic (Z‑test)

• Assume normal population (N(\mu,\sigma^2)) and known (\sigma).
• (Z=\frac{\bar{X}-\mu_0}{\sigma/\sqrt{n}})
• Under (H_0): (Z\sim N(0,1)).

4.3 p‑Value

• Two‑sided: (p=2[1-\Phi(|Z|)]).
• One‑sided (e.g., (H_0:\mu\le\mu_0) vs (H_1:\mu>\mu_0)): (p=1-\Phi(Z)).
• (\Phi) = standard normal CDF.

4.4 Decision Rule

• Significance level (\alpha) (default (0.05)).
• Reject (H_0) if (p<\alpha); otherwise fail to reject.

4.5 Power & Sample Size

• Power (=1-\beta) (probability of correctly rejecting (H_0)).
• For desired power, solve for (n) using the non‑centrality parameter derived from (\mu_1-\mu_0).

---

5. Confidence Intervals

• For (\mu) with known (\sigma):
  [ \bar{X}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}} ]
• Interpretation: 95% of such intervals will contain the true (\mu) if many samples are taken.

---

6. Practical Workflow for Algorithm Comparison

1. Collect Samples from both algorithms.
2. Compute (\bar{X}), (\bar{Y}), and estimated variances.
3. Select Test:
   • If (\sigma) known → Z‑test.
   • If (\sigma) unknown → t‑test (not detailed in slides but implied).
4. Calculate test statistic, p‑value.
5. Decide using (\alpha).
6. Report confidence interval for the difference (\bar{X}-\bar{Y}).

---

7. Quiz Details

• Date: Thursday, March 19th (beginning of class).
• Coverage: All material from the first lecture through today’s lecture, including Probability Theory and Statistics.
• Timing: Arrive on time; quiz starts immediately.

---