[kalman] update RV notation

[longye-tian]

Hi @mmcky

This PR updates the RV notation following #898 .

In particular, this extends the new uppercase/lowercase convention to the Implementation section and Exercises:

• use $X_t$ and $Y_t$ for random state and signal variables;
• keep lowercase notation for realizations, density arguments, filter estimates, and code variables;
• clarify that the Exercise 3 plotted errors use simulated realizations.

Best,
Longye

[mmcky]

Reviewed — the uppercase/lowercase RV convention is applied correctly and consistently, and the new x[:, t] is the realized value of $X_t$ note nicely bridges the math and the Ex3 code.

I pushed a small follow-up commit (dc7ca23) updating the two # Constant value of state x_t code comments to X_t to match. LGTM once CI is green.

[github-actions]

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Commit: 11bf337

📚 Changed Lectures

• kalman

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[Copilot]

Pull request overview

This PR updates the Kalman filter lecture (lectures/kalman.md) to consistently apply the “uppercase = random variable / lowercase = realization” convention across the Implementation section and Exercises, aligning with the notation introduced earlier in the lecture and requested in #898.

Changes:

• Updated the LinearStateSpace model equations in the Implementation section to use $X_t$ / $Y_t$ for random state and signal variables.
• Revised Exercises 1–4 text and code comments to use $X_t$ for random states while keeping lowercase for realizations/estimates/code variables.
• Clarified in Exercise 3 that the plotted squared errors are based on simulated realizations (and added an explicit note tying x[:, t] to the realized $X_t$).

[longye-tian]

Many thanks @mmcky for the update and I've addressed the comments from Copilot as well.

Best,
Longye