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Items / Reference templates / BP010

REF-BP010-1

C05 expected value · C05.1 expectation from pmf · target easy, CD2 · split dev · reference items carry their template's native demand and difficulty

Pending

A discrete random variable X has the distribution P(X = 0) = 0.12; P(X = 1) = 0.5; P(X = 4) = 0.12; P(X = 7) = 0.26. What is the expected value E[X]?

  1. A7.32M-SD-VS-VAR
  2. B1MISC-OTHER
  3. C2.8Key
  4. D3MISC-OTHER

Explanation (as written by the item's author)

E[X] = sum of x * P(X = x) = 2.8.

Distractors and the misconception each claims to execute
OptionCodeCatalogue meaningAuthor's rationale
AM-SD-VS-VARReports the variance where the standard deviation was asked, or vice versa.Reports the variance rather than the mean.
BMISC-OTHERDistractor is not attributable to any catalogued misconception (itself a quality signal).Reports the most probable value (the mode).
C (key)KEYProbability-weighted mean of the values.
DMISC-OTHERDistractor is not attributable to any catalogued misconception (itself a quality signal).Averages the values without weighting by probability.

Quality flags · 1 fired

  • inconsistent numeric format (style) firedNumeric options mix fractions, decimals and percentages, or use differing decimal precision.
  • numeric middle key (info) firedOptions are numeric and the key is one of the two middle values. Informational; tested at set level against the 0.5 chance rate.
  • duplicate options (integrity) not firedTwo options are identical after normalisation.
  • numerically equal options (integrity) not firedTwo options denote the same number (e.g. 0.5 and 1/2 and 50%).
  • missing options (integrity) not firedFewer than four non-empty options.
  • key out of range (integrity) not firedThe keyed answer is a probability outside [0, 1] (or a percentage > 100).
  • all none of above (cue) not firedAn option is 'all/none of the above' or 'both A and B'.
  • key uniquely longest (cue) not firedThe key is the longest option by at least 20% (length cue).
  • key uniquely shortest (cue) not firedThe key is the shortest option by at least 20%.
  • distractor out of range (cue) not firedA distractor is a probability outside [0, 1]; it can be eliminated without the construct.
  • absolute terms in distractors (cue) not firedDistractors (only) use absolute terms such as always/never/must/proves, a known cue that they are wrong.
  • stem option overlap cue (cue) not firedThe key alone repeats a content word from the stem.
  • grammatical mismatch (cue) not firedThe stem ends in 'a'/'an' and some options do not agree with it.
  • key is hedged only (cue) not firedOnly the key contains hedging language (may/might/plausibly/likely).
  • negative stem (style) not firedStem uses NOT/EXCEPT/LEAST phrasing.
  • overlong stem (style) not firedStem exceeds 110 words.
  • unit mismatch (style) not firedOptions carry different units.
  • option length imbalance (style) not firedLongest option is more than 2.5x the shortest.
  • rationale value mismatch (metadata) not firedA distractor's stated rationale computes a value ('= x') that is not the option it is attached to: the distractor does not actually execute the misconception it claims.

Validator output

PoT not yet run for this item.

Answering probes

Probes not yet run for this item.

Shortcut results

Construct-free heuristics (probability on the key; chance 0.25)
HeuristicP(key)
longest option0.00
shortest option0.00
convergence0.25
stem overlap0.25
numeric middle0.50
hedged option0.25
position prior1.00

Provenance and generation trace

Method
reference_template / —
Model
none
Source
parameterised template T-C05.1-pmf-mean, written by the AssessAI build agent (Claude); no external item bank; not human-reviewed
Prompt hash
—
LLM calls · cost
0 · $0.00000 · 0.0 s
Parser notes
none