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中文题目A machine has two critical components in series: component A with lifetime $\operatorname{Exp}(3)$ and component B with lifetime $\operatorname{Exp}(5)$, independent of each other. When either fails, the entire machine stops, the failed component is replaced (cost $\$20$ for A, $
打开 →Evaluate the infinite series sum_(n=1)^inf n * (1/2)^n.
打开 →For which of the following exponents p does the series sum_(n=1)^inf 1/n^p converge: p = 1/2, p = 1, p = 3/2, p = 2? List all values for which it converges.
打开 →Evaluate the infinite series sum_(n=1)^inf 1 / (n(n+1)).
打开 →Evaluate the infinite series sum_(n=1)^inf 1 / [(4n+1)(4n+1+4)].
打开 →Let a_n repeat the block [1, 0, 2] forever. Evaluate sum_(n=1)^inf a_n * (1/3)^n.
打开 →Let a_n repeat the block [2, -1] forever. Evaluate sum_(n=1)^inf a_n * (1/4)^n.
打开 →Let a_n repeat the block [1, 2, 0, 1] forever. Evaluate sum_(n=1)^inf a_n * (1/2)^n.
打开 →Let a_n repeat the block [4, -2, 1] forever. Evaluate sum_(n=1)^inf a_n * (1/6)^n.
打开 →Why is the real principle in time-series CV 'never train on information from the future' rather than 'always use a particular fold geometry'?
打开 →Evaluate the infinite series sum_(n=0)^inf 5 * (-2/3)^n.
打开 →What null hypothesis does the Ljung-Box test target in a return series, and what practical concern does rejection raise?
打开 →Determine the radius of convergence of the power series sum_(n=1)^inf (3^n / n) x^n.
打开 →Why can regressing one drifting series on another produce a large R-squared and tiny p-values even when there is no economic relation?
打开 →Find the value of the partial sum S_N = sum_(n=1)^N ln(1 + 1/n), and state whether the full series converges.
打开 →A sequence satisfies x_(n+1) = 1/2 x_n + c_n, where c_n = 1 on odd n and c_n = 3 on even n. Assuming any starting value x_1, what are the limits of the even and odd subsequences?
打开 →For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.6 and Var(e_t) = 1, what is the h = 3 step forecast error variance?
打开 →For the AR(1) model X_t = phi X_(t-1) + e_t with phi = 0.5 and Var(e_t) = 2.25, what is the h = 4 step forecast error variance?
打开 →A signal follows X_t = 0 + 0.6 X_(t-1) + e_t with Var(e_t) = 2 and current value X_t = 10. What is the h = 3 step forecast E[X_(t+3) | X_t]?
打开 →A signal follows X_t = 4 + 0.7 X_(t-1) + e_t with Var(e_t) = 1.5 and current value X_t = 8. What is the h = 2 step forecast E[X_(t+2) | X_t]?
打开 →A signal follows X_t = -1 + 0.8 X_(t-1) + e_t with Var(e_t) = 1 and current value X_t = 3. What is the h = 4 step forecast E[X_(t+4) | X_t]?
打开 →A GARCH(1,1) reduces to ARCH(1) when $\beta=0$: $h_t=\omega+\alpha r_{t-1}^2$. With $\omega=0.7$ and $\alpha=0.3$, compute the unconditional variance $\bar h$ as a decimal.
打开 →You observe the diagnostic statement: ACF tails off geometrically, PACF cuts after lag 1. What is the correct modeling conclusion?
打开 →You observe the diagnostic statement: ACF cuts after lag 1, PACF tails off. What is the correct modeling conclusion?
打开 →You observe the diagnostic statement: Both ACF and PACF tail off. What is the correct modeling conclusion?
打开 →You observe the diagnostic statement: AIC prefers ARMA(2,1) but BIC prefers ARMA(1,1). What is the correct modeling conclusion?
打开 →You observe the diagnostic statement: (1-0.5L) X_t = (1-0.5L) e_t. What is the correct modeling conclusion?
打开 →An expanding walk-forward starts with 12 months of training and then advances by 6 months for each of 5 complete test folds. What is the average training-window length used across the 5 folds?
打开 →Find the coefficient of x^10 in 1/((1-x^2)^2(1-x^3)).
打开 →Find the coefficient of x^8 in 1/((1-x)^2(1-x^3)).
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