Independent random variables

From equation 10.24 we see that is 1/A times the sum of Np independent random variables, each of which is equal to either - -1 or —1. is thus... 

One important consequence of this definition is that if lt - are statistically independent random variables then all groups of events of the form x in Aj, 2 in A2, , n in An are statistically independent.38 This property greatly simplifies the calculation of probabilities of events associated with independent random variables. 

Perhaps the most important property of statistically independent random variables is embodied in the following, easily verified, formula that is valid when - are statistically independent.39... 

Equation (3-179) states the quite remarkable result that the expectation of a product of statistically independent random variables is equal to the product of their individual expectations. 

Sums of Independent Random Variables.—Sums of statistically independent random variables play a very important role in the theory of random processes. The reason for this is twofold sums of statistically independent random variables turn out to have some rather remarkable mathematical properties and, moreover, many physical quantities, such as thermal noise voltages or measurement fluctuations, can be usefully thought of as being sums of a large number of small, presumably independent quantities. Accordingly, this section will be devoted to a brief discussion of some of the more important properties of sums of independent random variables. 

The last equation establishes the important result that the variance of a sum of statistically independent random variables is the sum of the variances of the summands. 

Another important result states that the characteristic function of a sum of statistically independent random variables is the product of the characteristic functions of the individual summands. The reader should compare this statement with the deceptively similar sounding one made on page 154, and carefully note the difference between the two. The proof of this statement is a simple calculation... 

Our next result concerns the central limit theorem, which places in evidence the remarkable behavior of the distribution function of when n is a large number. We shall now state and sketch the proof of a version of the central limit theorem that is pertinent to sums of identically distributed [p0i(x) = p01(a ), i — 1,2, ], statistically independent random variables. To simplify the statement of the theorem, we shall introduce the normalized sum s defined by... 

The Central Limit Theorem.—If 4>i,4>a, we identically distributed, statistically independent random variables having finite mean and variance, then... 

The central limit theorem thus states the remarkable fact that the distribution function of the normalized sum of identically distributed, statistically independent random variables approaches the gaussian distribution function as the number of summands approaches infinity—... 

H. Cramer, Mathematical Methods of Statistics, Princeton University Press, 1954 B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley Publishing Co., Inc., Cambridge, Mass., 1954 M. Lo ve, Probability Theory9 D. Van Nostrand Co., Inc., Princeton, N.J., 1955. 

Thus, the self information of a sequence of N symbols from a discrete memoryless source is the sum of N independent random variables, namely the self informations of the individual symbols. An immediate consequence of this is that the entropy of the sequence, ff(Ujr), is the average value of a sum of random variables so that... 

This is the average of the product of the N independent random variables, exp ( ). For independent random variables, the average of a product is equal to the product of the averages. Thus... 

Recall now that the letters in xx are chosen independently with the probability distribution p = (Pi, , > ) and when xx is sent the output is governed by the transition probabilities Pr( i). Thus, each of the terms d( ln,pn) in Eq. (4-123) is an independent random variable with the moment generating function... 

The random variable p with the probability density shown above is the sum of 2 independent random variables, namely the input a> with variance Aa2 and the noise with variance a2. Thus, Pr( y) satisfies... 

Summational invariants, 20 Sums of independent random variables, 155... 

There are two statistical assumptions made regarding the valid application of mathematical models used to describe data. The first assumption is that row and column effects are additive. The first assumption is met by the nature of the smdy design, since the regression is a series of X, Y pairs distributed through time. The second assumption is that residuals are independent, random variables, and that they are normally distributed about the mean. Based on the literature, the second assumption is typically ignored when researchers apply equations to describe data. Rather, the correlation coefficient (r) is typically used to determine goodness of fit. However, this approach is not valid for determining whether the function or model properly described the data. 

The proof that the variance of the sum of two terms is equal to the sum of the variances of the individual terms is a standard derivation in Statistics, but since most chemists are not familiar with it we present it in the Appendix. Having proven that theorem, and noting that AEs and AEr are independent random variables, they are uncorrelated and we can apply that theorem to show that the variance of AT is ... 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

The use of the semi-invariant expansion depends on the observation that if xx and x2 are independent random variables for which... 

Another way of handling changes of variables is through the moment generating function. If Z is the sum of two independent random variables X and Y, integration of the two variables under the integral can be carried out independently, hence... 

X is a normal random variable with mean p and variance a2. Given the set of samples of m observations with mean X and variance S2, which will be treated as independent random variables, show that the ratio... 

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. 

I> = 8c0ifi, and where I is the time spent at site i. When a random variable is defined as the sum of several independent random variables, its probability distribution is the convolution product of the distributions of the terms of the... 

If independent random variables x and y are uniformly distributed on a line, then their linear combination z = ax+Py is also uniformly distributed on a line. (Indeed, vector (x, y) is uniformly distributed on a plane (by definition), a set z>y is a half-plane, the correspondent probability is Vi.) This is a simple, but useful stability property. We shall use this result in the following form. If independent random variables are log-uniformly distributed on a line, then the... 

We first note that Eq. (29), which we derived by taking the limit x — 0 of the result (26) for general x, can also be obtained by a more direct route. In the limit x —> 0, the sizes of particles in the smaller system become independent random variables drawn from n (a) the second phase can be viewed as a reservoir to which the small phase is connected. One writes the moment generating function for V(m) in the small phase as a product of xN independent moment generating functions of n (o) and then evaluates the integral over V m) by a saddle point method [36]. 

B. V. Gdenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, MA, 1954. 

This formulation for the N-dimensional distance is directly related to the chi distribution. The chi distribution or % distribution is a probability distribution that describes the variation from the mean value of the normalized distance of a set of continuous independent random variables that each has a normal distribution. More formally if Xv Xv. .., XN are a set of N continuous independent random variables, where each X. has a normal distribution, then the random variable, Y, given by ... 

Let E[GRj be the expected value of the geometry factor G multiplied by the reflectance R,. Both can be considered as independent random variables, as there is no correlation between the shape and the color of an object. Let us assume that the reflectances are uniformly distributed, i.e. many different colors are present in the scene and each color is equally likely. Therefore, the reflectance can be considered to be a random variable drawn from the range [0, 1]. We obtain (Johnson and Bhattacharyya 2001)... 

The sum of identically distributed independent random variables is normally distributed for large sample size regardless of the probability distribution of the population of the random variable,... 

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 Erlang distributions used as retention-time distributions fi (a) have interesting mathematical properties considerably simplifying the modeling. For the Erlang distribution, it is well known that if v independent random variables Z, are distributed according to the exponential distribution... 

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