(1) Determine general relationships for the second, third, and fourth central moments (variance \(= \sigma^2\), skewness \(= S\), and kurtosis \(= K\)) of the random variable \(u\) in terms of its first four ordinary moments: \[ \overline{u},\quad \overline{u^2},\quad \overline{u^3},\quad \text{and} \quad \overline{u^4} \]
(2) Calculate the mean, mean square, variance, root-mean-square value, and standard deviation of the periodic time series \[ u(t) = \overline{U} + U_0 \cos(\omega t) \] where \(\overline{U}\), \(U_0\), and \(\omega\) are positive real constants.
(3) Show that the autocorrelation function \[ \overline{u(t)\,u(t+\tau)} \] of a periodic series\(u = U \cos(\omega t)\) is itself periodic.
(4) If \(u(t)\) is a stationary random signal, show that \(u(t)\) and \(\frac{du(t)}{dt}\) are uncorrelated.
(5) A mass of 10 kg of water is stirred by a mixer. After one hour of stirring, the temperature of the water rises by 1.0°C. What is the power output of the mixer in watts? What is the size \(\eta\) of the dissipating eddies?
1 This problem set contains 6 problems, and your final score will be based on the 4 problems on which you earned the highest scores.