This homework gives you some experience at algebraic manipulation of POMP models by deriving the prediction, filtering and smoothing formulas in Chapter 10.
The calculations are all applications of basic definitions (such as the Markov property) and basic identities for joint, conditional and marginal probability density functions. The goal is to check carefully how the formulas follow from these properties, so please explain this explicitly in your solutions. The hints for Exercises 10.2 and 10.4 in the notes may be useful.
The homework can be handwritten and scanned to pdf, however it is recommended to use Latex in Rmarkdown and submit as html. If you are relatively unfamiliar with Latex and Rmarkdown, this will take more time but it is a worthwhile exercise. As usual, you are welcome to use the Latex from the notes as a source, if you like.
Question 5.1. Derive the identity [P2].
Question 5.2. Derive the prediction formula, [P4].
Question 5.3. Derive the filtering formulas [P5] and [P6].
Question 5.4. Derive the backward recursion formulas [P7] and [P8].
Question 5.5. Derive the smoothing formula [P9].
Question 5.6. Explain which parts of your responses above made use of a source, meaning anything or anyone you consulted (including your class group, or other classmates, or online solutions to previous courses) to help you write or check your answers. All sources are permitted, but you are expected to explain clearly what is, and is not, your own original contribution, as discussed in the syllabus.