Terms Of Quadratures Research Articles

Joint Sampling Distribution of the Mean and Standard Deviation for Probability Density Functions of Doubly Infinite Range

The joint sampling distribution of $\bar{x}$ and $S$ is derived in integral form for probability density functions of doubly infinite range. This derivation is effected through the use of a transformation which transforms the sample probability element $f(x_1)f(x_2) \cdots f(x_n) dx_1 dx_2 \cdots dx_n$ into the element $f(x_1)f(x_2) \cdots f(x_{n - 2})f((n\bar{x} - \sum^{n - 2}_1 x_i \pm - \sum^{n - 2}_1 x_i \mp \Omega_1)/2) |J| \cdot dx_1 dx_2 \cdots dx_{n - 2}d\bar{x} dS,$ where $\bar{x} = (1/n) \sum^n_1 x_i, S^2 = (1/n) \sum^n_1(x_i - \bar{x})^2$, and $J$ is the Jacobian of the transformation. Bounds on $x_{n - r}, r = 2, 3, \cdots, n - 1$, are established in terms of $\bar{x}, S$, and $x_{n - r -j}, j = 1, 2, \cdots, n - r - 1$. The probability element $f(x)_1f(x_2) \cdots f(x_{n - 2})f((n\bar{x} - \sum^{n - 2}_1 x_i \pm \Omega_1)/2)f((n\bar{x} - \sum^{n - 2}_1 x_i \mp \Omega_1)/2)|J|dx_1 dx_2 \cdots dx_{n - 2}d\bar{x} dS$ must then be integrated with respect to $x_{n - r}, r = 2, 3, \cdots, n - 1$, between these limits to obtain $F(\bar{x},S)d\bar{x}dS$, the joint probability element of $\bar{x}$ and $S$. These limits of integration of $x_{n - r}, r = 2, 3, \cdots, n - 1$ enable one to express $F(\bar{x}, S)$ in terms of quadratures when $f(x)$ is any probability density function of doubly infinite range. To illustrate the method, $F(\bar{x}, S)$ is obtained when $f(x)$ is the normal probability density function.

Analysis of a Sampling Servo Mechanism

An analysis of a sampling servo mechanism with an error-clamping device and linear forward and return paths is given. The method leads to a determination of the continuous output of the system in terms of quadratures and gives explicitly the value of the output at the discrete sampling instants. Frequently this is all that is needed in order to study the output variation with time. The question of stability is discussed and a criterion for testing the stability of the system is given in terms of the system parameters.