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QUIZ 5: Filtering

Due date: 2026-06-13 23:59.
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Question 1: Deterministic state flow

Let 𝑋 = ℝ be the state space. Let 𝑓 : 𝑋 β†’ 𝑋 be the deterministic state flow defined as 𝑓(π‘₯) = 2π‘₯ + 1.
Let
X_{k+1}(X_k) = \{x_{k+1} \in X \mid x_k \in X_k \text{ and } x_{k+1} = f(x_k) \}. \\\
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Question 2: Nondeterministic state flow

Let 𝑋 = ℝ be the state space. Let 𝑓 : 𝑋 β†’ pow(𝑋) be the nondeterministic state flow defined as:
f(x) = \{x + d \in X \;|\; d \in [-1,1] \}. \\\
Let
X_{k+1}(X_k) = \{x_{k+1} \in X \mid x_k \in X_k \text{ and } x_{k+1} \in f(x_k) \}. \\\
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Question 3: Nondeterministic temporal filtering

Consider the same setup as in the previous problem, but now include a sensor mapping β„Ž : 𝑋 β†’ π‘Œ defined by:
y = \lfloor x + \tfrac{1}{2} \rfloor, \\\
in which βŒŠΒ·βŒ‹ denotes the floor function (equivalent to the floor function in Python).
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Question 4: Discrete probability statements

Select ALL correct statements:
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Question 5: More discrete probability statements

Select ALL correct statements:
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Question 6: Gaussians and sampling

Let 𝑆 βŠ‚ ℝ be a collection of 10,000 distinct samples that have mean πœ‡ and variance 𝜎².
Make a set 𝑆₁ = { π‘₯ ∈ ℝ ∣ π‘₯ βˆ’ 1 ∈ 𝑆 } by adding 1 to all of the samples. Make another set 𝑆₂ = { π‘₯ ∈ ℝ ∣ π‘₯ / 2 ∈ 𝑆 } by multiplying all of the samples by 2. Let πœ‡α΅’ and 𝜎ᡒ² denote the mean and variance, respectively, of 𝑆ᡒ for 𝑖 = 1, 2.
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Question 7: Probabilistic filters

Select all correct statements:
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Question 8: More probabilistic filters

Select all correct statements:
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Authors

Steven M. LaValle.
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