Medium · Statistics & Statistical Inference · Numerical · ~6 min · #89

Best Guess from a Noisy Signal

Real interview · Citadel, Susquehanna (SIG)

$X \sim N(0, 1)$ and independent noise $Z \sim N(0, 3)$ , where 3 is the variance. You observe $Y = X + Z = 2$ . What is $\mathbb{E}[X \mid Y = 2]$ ?

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Worked solution

Give the problem an honest attempt first, then reveal the full explanation.

Reported in interviews

Citadel· Quantitative ResearcherGlassdoor interview reviews
Reported for general noise variance, asking for the full distribution of X given Y.
Susquehanna (SIG)Glassdoor interview question
Reported as the best guess for x given known standard deviations and the sum.

Candidate-reported and unverified; the firm may phrase it differently or no longer ask it. The wording here is our own paraphrase, and the solution was written and checked independently of any answers posted at the source.

Concepts

Conditional GaussiansShrinkage

Relevant for

Quant ResearcherDesk Quant