Testwiki:Reference desk/Archives/Mathematics/2012 August 17

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August 17

Stationary problems in time series

Having issues finding the mean, covariance and variance of the following: (where Ut is i.i.d with mean 0 and variance sigma squared)

1. ) $y_t=y_1{e}^{\beta t}{e}^{U_t}$
2. ) $y_t={\beta }_0+{\beta }_1{x}_t+U_t$, where $x_t={\alpha }_0+V_t$ , $V_t$ is i.i.d with mean 0 and variance σ squared subscript v
3. ) $\mathrm{\Delta }{y}_t={\alpha }_0+{\alpha }_1\mathrm{\Delta }D{T}_b+U_t$, Where $D{T}_b$ is equal to 0 if t is less than or equal to $T_t$ and $t-T_b$ if $t$ is $>T_b$

— Preceding unsigned comment added by 203.28.240.65 (talk • contribs) 22:48, 16 August 2012

We'll only help with homework questions if you show that you have attempted it yourself first. How far have you managed to get? Also, can you clarify the question: covariance between what and what? --Tango (talk) 13:36, 18 August 2012 (UTC)

I can get to working out what the formulas would be, and I think you just assume linear independence between the error terms and the independent variable, allowing the substitutions to be easier. The covariance is meant to be between the y_t or Delta y_t and the independent variable.

— Preceding unsigned comment added by 203.28.240.65 (talk • contribs) 10:11, 19 August 2012‎

Integration of sinh(ax)/(e^(bx)-1) dx from 0 to ∞

How can we reach the following integration result?

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{\sinh (ax)}{e^{bx}-1}dx=\frac{1}{2a}-\frac{\pi }{2b}\cot (\frac{a}{b}\pi )$

I am not certain if solution method can be done using Residue theorem. If so then shall we make it through a quarter circle and how?--Almuhammedi (talk) 03:36, 17 August 2012 (UTC)

See here. Count Iblis (talk) 16:42, 17 August 2012 (UTC)