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Video status: Related supplementary video included; the original lecture recording has not been verified. Source details
This is article 7 in Reading Stanford CS109, covering Summer 2026 Lecture 6: Moments (Expectation) on June 30 with Chris Gregg. Its Summer agenda comes from the schedule, worksheet, answer key, and LLM guide. The /spr26 reader is a shared, Spring-dated concept reference rather than evidence of the Summer lecture. Canvas video is inaccessible and is not reconstructed.
From a distribution to an operational summary
Expectation is the probability-weighted average,
E[X] = Σx x P(X=x)and need not be attainable. A fair die has expectation 3.5 although no face shows 3.5.
Problem 1: reconnecting the binomial PMF
For five four-option questions guessed independently, X~Bin(5,.25) and
P(X=k) = C(5,k)(.25)^k(.75)^(5-k)Exactly two substitutes k=2; at least one is 1-.75^5. This separates a fixed-trial success count from a waiting-time variable.
Problem 2: naming four classic variables
A single ad click is Bernoulli(.01); dates until the first partner are geometric(.2); attempts until Shaq’s third make are negative binomial with r=3 and p=.53; heads in twenty fair flips are Bin(20,.5). Identify what X counts and whether trials or required successes determine stopping.
Problem 3: sampling a class versus sampling a student
Choosing one of classes sized 5, 10, and 150 uniformly gives mean 55. Choosing one of 165 students uniformly weights classes by enrollment, giving (5²+10²+150²)/165≈137.1. The larger student-experienced average is size-biased sampling, not contradictory arithmetic.
Problem 4: linearity and LOTUS
For die result X and winnings W=2X-1, linearity gives E[W]=2E[X]-1=6. LOTUS handles nonlinear transformations:
E[g(X)] = Σx g(x)P(X=x)Thus E[X²]=91/6, while E[X]²=12.25; averaging then squaring differs from squaring then averaging. That contrast prepares the variance topic without claiming it as this lecture's agenda.
Problem 5: expectations of classic variables
The formulas are E[Bern(p)]=p, E[Bin(n,p)]=np, and E[Geo(p)]=1/p. Shaq expects 265 makes in 500 attempts at .53; improving by ten percentage points adds 50 expected makes. Geometric(.2) has mean waiting time five, a long-run average rather than a fifth-trial guarantee.
Problem 6: daycare revenue and piecewise cost
With six babies independently attending with probability 5/6, X~Bin(6,5/6) and high attendance is P(X=5)+P(X=6). Revenue is linear:
E[R] = 50E[X] = 50 × 6 × 5/6 = 250Staffing cost is piecewise: 200P(X≤4)+400P(X≥5). This contrasts a linear transform with a function requiring event-weighted LOTUS reasoning.
Problem 7: linearity without independence
Let Xi indicate whether position i contains its sorted card. Each is Bernoulli(1/52), though the indicators are dependent. Linearity still gives
E[X] = E[Σ Xi] = Σ E[Xi] = 52 × 1/52 = 1so a random permutation has one fixed point on average. The challenge derives E[Bin(n,p)]=np by writing the count as a sum of n Bernoulli indicators—no binomial-PMF sum required.
Material gaps
- This guide covers all seven worksheet problems and the optional challenge without inventing inaccessible classroom examples.
- Recordings are Canvas-gated and were not used.
Course video sources
Verified the official Summer 2026 archive: the course is recorded and its syllabus says recordings are available through Canvas. The archived homepage's Videos link redirects to Stanford Canvas login and requires course access, so there is no public Summer 2026 player. The 2022 Stanford Online public recording(s) below partly overlap this lecture's topic (see the content check below) but come from a different offering with different lecture numbering; they are supplementary material, not Summer 2026 lecture recordings.
Original videos: Stanford CS109 Probability for Computer Scientists I Variance Bernoulli Binomial I 2022 I Lecture 7
Official sources:
- CS109 Summer 2026 — Canvas lecture recordings (login required)
- CS109 Summer 2026 — archived official schedule
- CS109 Summer 2026 — archived syllabus and recording policy
- Stanford Online CS109 2022 public playlist (different offering, supplementary)
Checked: 2026-10-10.
Content check: verified against the video transcript (2026-10-10): what was checked is how the video topic relates to this lecture. This is 2022 Lecture 7 (Variance, Bernoulli, Binomial): it opens by recapping linearity of expectation and LOTUS, then covers Bernoulli and binomial, and only introduces variance at the end. It is adjacent to and partly overlaps this lecture (expectation, LOTUS, linearity) but weighs variance and binomial more heavily, so it is a related supplement, not the same lecture.
Update Log
- 2026-10-10: Added explicit video status and checked recording sources and access notes.
- 2026-10-10: Rechecked video status. Switched to related supplementary video: embedded the 2022 Stanford Online public recording on the same topic; the Summer 2026 original is behind Canvas sign-in and could not be verified.
- 2026-10-10: Checked the video content against its transcript. The video only partly overlaps this lecture (expectation recap); its main body is Bernoulli/binomial/variance, now stated in the sources section.
References
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