This homework gives you some experience at algebraic manipulation of POMP models by deriving the prediction, filtering and smoothing formulas in Chapter 9 of the notes.
The calculations are all applications of basic identities for joint, conditional and marginal probability density functions. You may follow the hints in Section 9.2.5 of the notes.
Please say explicitly when you use the Markov property. Also, write explicitly when and how you use basic identities such as those provided in Section 9.2.5.
You are expected to turn in your homework on paper, in class, since that is the easiest way to do the math. If you want to Latex your solution, you are welcome to do it.
Question 6.1. Derive the identity [MP2].
Question 6.2. Derive the prediction formula, [MP4].
Question 6.3. Derive the filtering formulas [MP5] and [MP6].
Question 6.4. Derive the backward recursion formulas [MP8] and [MP9].
Question 6.5. Derive the smoothing formula [MP10].