Two Random Variables

Normal distribution and that they are statistieally independent. The problem is governed, in faet, by the amount of negative elearanee. This translates into a differenee between the mean of the two random variables, /i, and i, (see Figure 4). The mean of this differenee distribution, /i, is given by ... 

The expectation symbol E obeys the same rules of manipulation, Eq. (3-40), as in the one-dimensional case. The only additional comment needed here is that the addition rule holds even when the two random variables concerned are defined with respect to different sets of r s. The proof of this fact is immediate when the various expectations involved are written as time averages. 

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... 

Having two random variables, x and y, e.g. values measured independently from each other and each of them being normally distributed, then the following questions may be of interest ... 

Just as in everyday life, in statistics a relation is a pair-wise interaction. Suppose we have two random variables, ga and gb (e.g., one can think of an axial S = 1/2 system with gN and g ). The g-value is a random variable and a function of two other random variables g = f(ga, gb). Each random variable is distributed according to its own, say, gaussian distribution with a mean and a standard deviation, for ga, for example, (g,) and oa. The standard deviation is a measure of how much a random variable can deviate from its mean, either in a positive or negative direction. The standard deviation itself is a positive number as it is defined as the square root of the variance ol. The extent to which two random variables are related, that is, how much their individual variation is intertwined, is then expressed in their covariance Cab ... 

If two random variables are nncorrelated, then both their covariance Cab and their correlation coefficient rab are equal to zero. If two random variables are fully correlated, then the absolute value of their covariance is C,J = cacb, and the absolute value of their correlation coefficient is unity rab = 1. A key point to note for our EPR linewidth theory to be developed is that two fully correlated variables can be fully positively correlated rab = 1, or fully negatively correlated rab = -1. Of course, if two random variables are correlated to some extent, then 0 < Cab < oacb, and 0 < IrJ < 1. 

The difference between two random variables is itself a random variable, therefore we replace the terms (AE s - AE 0s) and (AEX - A (jr) in equation 41-3 with the equivalent, simpler terms AEs and AEr, respectively ... 

Hint this is most easily done by projecting each distribution into one dimension, where it becomes a Gaussian, and using the theorem from statistics that the variance of the difference (or sum) of two random variables is the sum of the individual variances. 

Here 4 is the target state vector at time index k and Wg contains two random variables which describe the unknown process error, which is assumed to be a Gaussian random variable with expectation zero and covariance matrix Q. In addition to the target dynamic model, a measurement equation is needed to implement the Kalman filter. This measurement equation maps the state vector t. to the measurement domain. In the next section different measurement equations are considered to handle various types of association strategies. 

The two random variables on the right-hand side are completely determined by the two-time PDF, /u,u (V, V x, t, r). Thus, the expected value of the time derivative can be defined by... 

An important concept is the marginal density function which will be better explained with the joint bivariate distribution of the two random variables X and Y and its density fXY(x, y). The marginal density function fxM(x) is the density function for X calculated upon integration of Y over its whole range of variation. If X and Y are defined over SR2, we get... 

Two random variables X and Y are independent if their joint density function fXY can be factored as a product of two density functions, each involving one variable, e.g.,... 

In the theory of probability the term correlation is normally applied to two random variables, in which case correlation means that the average of the product of two random variables X and Y is the product of their averages, i.e., X-Y)=(,XXY). Two independent random variables are necessarily uncorrelated. The reverse is usually not true. However, when the term correlation applies to events rather than to random variables, it becomes equivalent to dependence between the events. 

To study the relation between the number of features and the system quality, we may use one of standard measures that characterize the difference between two random variables. Assume that the scores ch and cw, related to comparisons between different and the identical eyes, are independent random variables. The decidability (or detectability) [A2] is defined as... 

The two random variables x andy are called independent or uncorrelated if... 

To illustrate the coefficient of variation, consider the following (extremely simple and artificial) example. Imagine tliat there are two random variables in an early therapeutic exploratory clinical trial. One random variable is pulse (ranging from 50 to 80) and the other is age, which in this case is pulse minus 20. We can see that, from this example, values of pulse and age are just as disperse, but what differs between them is the mean. Hence, when we calculate the standard deviation, one random variable will appear to have more or less dispersion, but, after re-scaling the standard deviation with the sample mean, the measure of dispersion is the same. 

Correlation is one of the most important concepts in statistics [Pearson, 1920], being a quantity that indicates the strength and direction of a linear relationship between two random variables... 

The average quantity in (7.118) can be viewed as an average of a product of two functions. If these were independent (in the sense of the independence of two random variables), one could rewrite this average as a product of two... 

In other words, if all the random variables are independent, the joint pdf can be factored into the product of the individual pdfs. For two random variables, Xi and X2, their joint distribution is determined from their joint cdf... 

An extension to the bivariate normal distribution is when there are more than two random variables under consideration and their joint distribution follows a multivariate normal (MVN) distribution. The pdf for the MVN distribution with p random variables can be written as... 

Suppose the joint distribution of the two random variables x andy is... 

We can see in Fig. 2.8(b) that high values of y tend to occur together with high values of x, and vice versa. In these cases, we say that the two random variables present a certain covariance, that is, a tendency to deviate conceitedly from their respective averages. We can obtain a numerical measure of the covariance from the products of the deviations (atj — Jc) and (jj — y) for each member of the sample. Since in this example the two deviations tend to have the same sign, be it positive or... 

To understand how the normal distribution arises, we must introduce another basic concept, that of independent random variables. Two random variables, X and Y, are said to be independent if... 

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