Conditional probability distributions

The conditional probability distribution function of the random variables fa, , fa given that the random variables fa, , fa+m have assumed the values xn+1, , xn+m respectively, can be defined, in most cases of interest to us, by means of the following procedure. To simplify the discussion, we shall only present the details of the derivation for the case of two random variables fa and fa. We begin by using the definition, Eq. (3-159), to write... 

One way to introduce the data values into the estimation algorithm is to consider the conditional probability distribution of the unknown P( c), given the N data values used to estimate it Denote this conditional cumulative distribution function (cdf) by ... 

Lso is the commutator with the electron spin Zeeman Hamiltonian (assuming isotropic g tensor, Hso = gS- Bo), Lrs = Lzfs (the sub-script RS stands for coupling of the rotational and spin parts of the composite lattice) is the commutator with the ZFS Hamiltonian and Lr = —ir, where is a stationary Markov operator describing the conditional probability distribution, P(QolQ, t), of the orientational degrees of freedom through ... 

Figure 9.1 The conditional probability distribution, P(4>hk). of the three-phase structure invariants, 4>hK having associated parameters Ahk with values of 0,1, 2,4, and 6. When A ss 0, all values of 4>jjk 3te equally likely, and no information useful for phase determination is available. However, the sum of the three phases for most invariants with A 6 is close to 0°, and an estimate of one phase can be made if the other two are known.

Using the estimation procedure described in Section 4.1 below, estimate the conditional probability distribution ftinction (pdf) of the DFT coefficients absolute value, given that they originate from a u type coefficients or an 1 type. 

The most telling form of result from this equation is the conditional probability distribution p(x, r, xo, To) dx- This is the probability of the system s being in the interval (x, X + dx) at time r, given that it was at 0 at time t0. p satisfies... 

The future development of the process taking into account the knowledge of the past can be analyzed by conditional probability distributions and densities, for example,... 

The problem considered here is the estimation of the state vector X (which contains the unknown parameters) from the observations of the vectors = [yo> yi.yk ] Because the collection of variables Y = (yoYi - -yk) jointly gaussian, we can estimate X by maximizing the likelihood of conditional probability distributions p(Xk/Yk), which are given by the values of conditional variables. Moreover, we can also search the estimate X, which minimizes the mean square error k = Xk — Xk. In both cases (maximum likelihood or least squares), the optimal estimate for the jointly gaussian variables is the conditional mean and the error in the estimate is the conventional covariance. 

The Cochran formula, Equation (9), estimates the triplet phase only exploiting the information contained in the three moduli hl, kU h+kl- The representation theory proposed by Giacovazzo " indicates how the information contained in all reciprocal space could be used to improve the Cochran s estimate of The conclusive conditional probability distribution has again a von Mises expression ... 

The conditional probability distribution / (x y) is defined so thatP(x y)dr is the probability that the value of x is in the interval x,..., x + dx given that the variable y takes the value y. From this definition it follows that... 

A statistical model of the noise, namely, a conditional probability distribution Py x of the attack channel. 

Figure 7.1. A causal graph for risk analysis. The model depicted in this figure can be formalized using a Bayesian network (Ricci et al. 2006) A probabilistic framework interprets the model described in this figure as a Bayesian belief network or causal graph model. Each variable with inward-pointing arrows is interpreted as a random variable with a conditional probability distribution that depends only on the values of the variables that point into it. The essence of this approach to modeling and evaluating uncertain risks is to sample successively from the (often conditional) distribution of each variable, given the values of its predecessors. Algorithms exist to identify and validate possible causal structures.

We have shown in Appendix A how these results can be obtained directly from the Langevin equation). Thus the conditional probability distribution of the velocities in the Ornstein-Uhlenbeck [11] process is... 

Figure 3. Conditional probability distributions for the valence electrons of the alkaline earth atoms in their ground states, for various values of the distance between the nucleus and one valence electron. The results are taken from calculations of Ref. 27.

Notice that two assumptions have been made normality of the responses and constant variance. The result is that the conditional distribution itself is normally distributed with mean 0O + (fix and variance joint distribution function at any level of X can be sliced and still have a normal distribution. Also, any conditional probability distribution function of Y has the same standard deviation after scaling the resulting probability distribution function to have an area of 1. 

The characterization of the spatial distribution of TCDD concentrations was performed by obtaining estimates of the conditional probability distribution function, I (x z ) for each of the six concentration cutpoints, zk> 6. These estimates... 

The conditional probability of an unknown TCDD concentration in an area being less than a threshold, T, can be found by simple linear interpolation from the estimated conditional probability distribution function. This interpolation is of the form ... 

The indicator approach to estimation of spatial distributions of TCDD concentrations consists in estimating first the conditional probability distribution of any unknown concentration. Estimates of the unknown concentration are then derived, together with their confidence intervals. This approach has several outstanding features ... 

Before defining a. Markov process a t), it is necessary to introduce conditional probability distribution functions 7) 02, 2 I 3) defined by... 

The orientation of a general anisotropic molecule (not necessarily a rigid rod) is given by the Euler angles O (a, fS, y) which specify the orientation of the molecular body-fixed axes with respect to the space-fixed axes (see Fig. 7.4.1).9 What is required is the conditional probability distribution KS(Q, t Do) which specifies the probability distribution for the molecule to have an orientation O at time t given thaPit had an orientation Qo at time 0. This conditional distribution must satisfy the initial condition... 

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