Stanford CS109 Lecture 9 | Normal Distribution: Standardization, Phi, and continuity correction

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This is article 10 in Reading Stanford CS109, covering Summer 2026 Lecture 9: The Normal Distribution on July 6 with Chris Gregg. Its Summer agenda follows the worksheet, answer key, LLM Learning Guide, and shared Spring-dated reader chapters on the Normal and binomial approximation. The Canvas recording is inaccessible, so spoken material is not reconstructed.

The original worksheet has two pages: P1–P3 are on page one, while P4–P7 and the challenge are on page two. No problem number is missing. P5 and the challenge are problem-set items deliberately hidden from the public answer key; this guide distinguishes its derivations from officially printed solutions.

P1: Review Exponential waiting time

The wait for a server's next request is T~Exp(0.5) hours:

P(T<2) = 1-e^(-0.5×2) = 1-e^-1 ≈ 0.632
E[T] = 1/0.5 = 2 hours

Given no request in the first three hours, memorylessness gives

P(T>5 | T>3) = P(T>2) = e^-1 ≈ 0.368

This closes the waiting-time model before the worksheet introduces the Normal distribution for measurement error, natural variation, and sums of many small effects.

P2: Normal parameters and standardization

For an exam score X~N(70,16), CS109's second parameter is variance σ²=16, so the mean is 70 and standard deviation is σ=4. Many software libraries take σ rather than σ², an interface trap the worksheet explicitly flags.

The z-score of 74 is

z = (74-70)/4 = 1
Z = (X-μ)/σ ~ N(0,1)

Thus 74 is one standard deviation above the mean. With Φ(z)=P(Z≤z) denoting the standard-normal CDF, P(X<74)=Φ(1)≈0.841. Standardization is an exact linear transformation of a Normal, not an approximation.

P3: Symmetry of Phi and intervals

The standard normal is symmetric around zero:

Φ(-a) = 1-Φ(a)

The probability within one standard deviation is

P(-1≤Z≤1) = Φ(1)-Φ(-1)
           = 2Φ(1)-1
           ≈ 0.683

This is the first part of the 68–95–99.7 rule. A right tail uses a complement: if Φ(1.31)=0.9049, then P(Z>1.31)=0.0951. Sketch the event before choosing Φ, 1-Φ, or a difference of two CDF values.

P4: A two-sided submarine-panel specification

Panel thickness is X~N(500,36) microns, hence σ=6. The specification accepts 490 through 510, whose z-scores are

z490 = (490-500)/6 ≈ -1.667
z510 = (510-500)/6 ≈  1.667

Therefore

P(490≤X≤510)
= Φ(1.667)-Φ(-1.667)
= 2Φ(1.667)-1
≈ 0.904

About 90.4% of panels meet the standard. The problem combines P2's standardization with P3's symmetry: convert each endpoint, then subtract the left CDF from the right.

P5: A Website Analytics right tail

Weekly visitors satisfy X~N(2200,52900), so σ=230. The probability of exceeding 2,000 is

P(X>2000)
= 1-Φ((2000-2200)/230)
= Φ(200/230)
≈ 0.808

The threshold lies below the mean, so the result should exceed one half—a useful directional check. P5 belongs to pset3 and is omitted from the public answer key; 0.808 is computed directly from the public prompt and the standard-normal CDF.

P6: Normal approximation to a large Binomial

A new design is tested on one million users. Under the no-effect assumption, each improves independently with probability 0.5, so X~Bin(10⁶,0.5). The approximating Normal has

μ = np = 500,000
σ² = np(1-p) = 250,000
σ = 500

The CEO endorses at X≥501,000. Integer 501,000 occupies a discrete bar beginning at 500,999.5 on the continuous curve, so continuity correction gives

P(X≥501,000)
≈ 1-Φ((500,999.5-500,000)/500)
= 1-Φ(1.999)
≈ 0.0228

Even with no real effect, sampling variation reaches the endorsement threshold about 2.3% of the time. Here p=0.5 and np(1-p)=250,000 is far above ten, making this a large-n, moderate-p Normal regime. Poisson is instead suited to large n with tiny p.

P7: What continuity correction corrects

For 100 fair flips, the heads count X~Bin(100,0.5) is approximated by N(50,25). The discrete event X=55 is a bar one unit wide, so preserve the entire bar:

P(X=55) ≈ P(54.5<Y<55.5)
        = Φ((55.5-50)/5)-Φ((54.5-50)/5)

The event X≤60 includes all of the 60 bar, moving the continuous boundary to 60.5:

P(X≤60) ≈ Φ((60.5-50)/5) = Φ(2.1)

Drawing integer bars is safer than memorizing signs: ≤k extends to k+0.5, ≥k begins at k-0.5, and =k occupies [k-0.5,k+0.5].

Challenge: A linear combination of independent Normals

Let independent X~N(1,2) and Y~N(1,2), with W=2X+Y. A linear combination of Normals remains Normal. Means use the coefficients directly, while variances use squared coefficients:

E[W] = 2E[X]+E[Y] = 3
Var(W) = 2²Var(X)+Var(Y) = 4×2+2 = 10
W ~ N(3,10)

Thus

P(W<5) = Φ((5-3)/√10) ≈ Φ(0.632) ≈ 0.736

Independence makes the covariance term zero. Without it, the variance would also contain 2ab Cov(X,Y). This challenge belongs to pset4 and is hidden from the public key; the result above is derived from the prompt.

How to use the LLM Learning Guide

The guide's six concepts are Normal parameters, standardization and Φ, symmetry and intervals, linear transformations and sums, Normal approximation to a binomial, and continuity correction. A useful test order is to distinguish variance from SD, sketch the event direction, and only then look up Φ. For an approximation, add one sentence explaining Normal rather than Poisson and draw the half-unit adjustment around discrete bars.

Material boundaries

• This guide covers worksheet P1–P7, the optional challenge, and all six LLM-guide concepts; there is no page-boundary numbering gap.
• P5 and the challenge are problem-set items deliberately omitted from the public answer key; this article derives them only from the public prompts.
• The Canvas recording is inaccessible, so no additional spoken examples or claims are inferred.
• The worksheet and guide are only two pages each. The short-material exception applies: complete problem coverage without generic padding. It remains draft: true pending independent review.

References

• Stanford CS109 Summer 2026 schedule
• Lecture 9: The Normal Distribution
• Lecture 9 worksheet
• Lecture 9 answer key
• Lecture 9 LLM Learning Guide
• Probability for Computer Science: Normal
• Probability for Computer Science: Normal approximation to Binomial