Chapter 4.1, Problem 122E

State and prove the Morton's inequality Theorem 1.1 (Markov's inequality) Suppose that E|X|" 0, and let x > 0. Then, E|X|" P(|X|> x) ≤ x"

(iii) If, in addition, X1, X2, ... Xn are identically distributed, then P(S|>x) ≤2 exp{-tx+nt²o}}.

Calculate the following limit lim N→X [en] + [en] + n + [en]

Solve the given symbolic initial value problem and sketch a graph of the solution. y"+y=38 (1-2); y(0) = 0, y'(0) = 2

Solve the following system of equations: 50x+20y=1800 10x+3y=300

5. State space models Consider the model T₁ = Tt−1 + €t S₁ = 0.8S-4+ Nt Y₁ = T₁ + S₁ + V₂ where (+) Y₁,..., Y. ~ WN(0,σ²), nt ~ WN(0,σ2), and (V) ~ WN(0,0). We observe data a. Write the model in the standard (matrix) form of a linear Gaussian state space model. b. Does lim+++∞ Var (St - St|n) exist? If so, what is its value? c. Does lim∞ Var(T₁ — Ît\n) exist? If so, what is its value?

2 P(x,y). kx²y X: 1,2 5.11273 Find k Find P(x/y) ③ Mxy Ng q oxy ว

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