Binomial distribution variance

Where f(x) is tlie probability of x successes in n performances. One can show that the expected value of the random variable X is np and its variance is npq. As a simple example of tlie binomial distribution, consider tlie probability distribution of tlie number of defectives in a sample of 5 items drawn with replacement from a lot of 1000 items, 50 of which are defective. Associate success with drawing a defective item from tlie lot. Tlien the result of each drawing can be classified success (defective item) or failure (non-defective item). The sample of items is drawn witli replacement (i.e., each item in tlie sample is relumed before tlie next is drawn from tlie lot tlierefore the probability of success remains constant at 0.05. Substituting in Eq. (20.5.2) tlie values n = 5, p = 0.05, and q = 0.95 yields... 

It would be of obvious interest to have a theoretically underpinned function that describes the observed frequency distribution shown in Fig. 1.9. A number of such distributions (symmetrical or skewed) are described in the statistical literature in full mathematical detail apart from the normal- and the f-distributions, none is used in analytical chemistry except under very special circumstances, e.g. the Poisson and the binomial distributions. Instrumental methods of analysis that have Powjon-distributed noise are optical and mass spectroscopy, for instance. For an introduction to parameter estimation under conditions of linked mean and variance, see Ref. 41. 

This result is obtained from the binomial distribution if we let p approach 0 and n approach infinity. In this case, the mean fx = p approaches a finite value. The variance of a Poisson distribution is given as cr = fx. 

There are many other distributions used in statistics besides the normal distribution. Common ones are the yl and the F-distributions (see later) and the binomial distribution. The binomial distribution involves binomial events, i.e. events for which there are only two possible outcomes (yes/no, success/failure). The binomial distribution is skewed to the right, and is characterised by two parameters n, the number of individuals in the sample (or repetitions of a trial), and n, the true probability of success for each individual or trial. The mean is n n and the variance is nn(l-n). The binomial test, based on the binomial distribution, can be used to make inferences about probabilities. If we toss a true coin a iarge number of times we expect the coin to faii heads up on 50% of the tosses. Suppose we toss the coin 10 times and get 7 heads, does this mean that the coin is biased. From a binomiai tabie we can find that P(x=7)=0.117 for n=10 and n=0.5. Since 0.117>0.05 (P=0.05 is the commoniy... 

Thus, we have a simplified distribution characterized by one parameter, xm compared to two parameters in the binomial distribution. The Poisson distribution is an asymmetric distribution as shown in Figure 18.24. Besides being a more tractable function to use, the Poisson distribution has certain important properties that we will use in analyzing radioactivity data. Let us consider a parameter, the variance, cr2, which expresses something about the width of the distribution of values about the mean, xm. For a set of N measurements, we can calculate cr2 as ... 

In developing a procedure for bacteriological testing of milk, samples were tested in an apparatus that includes two components bottles and kivets. All six combinations of two bottle types and three kivet types were tested ten times for each sample. The table contains data on the number of positive tests in each of ten testings. If we remember section 1.1.1 then the obtained values of positive tests are a random variable with the binomial distribution. For a correct application of the analysis of variance procedure, the results should be normally distributed. It is therefore possible to transform the obtained results by means of arcsine mathematical transformation for the purpose of example of three-way analysis of variance with no replications, no such transformations are necessary. The experiment results are given in the table ... 

Equation 7.4-1 shows that the distribution of the minor component in the samples depends both on the average concentration of the minor component, p, and on the size of the sample, n. This point becomes more evident by considering the variance of the binomial distribution... 

The more particles the samples contain, the narrower the distribution. In samples taken from true solutions, where the ultimate particles are molecules, the number of molecules in the smallest practical withdrawn sample is enormous, the variance will approach a value of zero, and the distribution will be virtually uniform. Figure 7.35 demonstrates the effect of the sample size on the shape of the binomial distribution. 

In practice, simpler though less reliable tests are used for evaluating the state of mixing. One of these involves calculating certain mixing indices that relate representative statistical parameters of the samples, such as the variance and mean, to the corresponding parameters of the binomial distribution. One such index is defined as follows ... 

The Binomial Distribution Consider a random mixture of minor particles in major particles of equal size. The fraction of minor particles in the mixture is p. We withdraw a large number of testing samples from the mixture, each containing exactly n particles, (a) Show that distribution of minor particles in the samples is given by Eq. 7.4-1. (b) Calculate the mean and the variance of the distribution. 

Assume that too independent units were introduced initially into the system with a transfer mechanism whose hazard rate h applies to all units in the experiment. The random movement of individual units in the heterogeneous process will result in a state probability p (t, h) depending on the specific h of all units in that experiment. Using the binomial distribution, the conditional expectation and variance are... 

The variance expression is composed of two terms m0ps (t) generalizes the variance of a standard binomial distribution and is attributable to the stochastic transfer mechanism (structural heterogeneity) and rn pp (f) reflects the random nature of h (functional heterogeneity). 

It should be noted that this Poisson distribution is still a discrete distribution. In radioactive decay, each atom can assume only one of two states disintegrated or intact. It is the fact that there is such a large number of atoms that decay follows the Poisson distribution, as a limiting case of the binomial distribution. The variance of the Poisson distribution (o2) is equal to the mean. That equality is the basis for the feet that the accuracy of radioactive measurements (or indeed any similar observation following a Poisson distribution) is proportional to the square root of the number of observations. 

In the Poisson and binomial distributions, the mean and variance are not independent quantities, and in the Poisson distribution they are equal. This is not an appropriate description of most measurements or observations, where the variance depends on the type of experiment. For example, a series of repeated weighings of an object will give an average value, but the spread of the observed values will depend on the quality and precision of the balance used. In other words, the mean and variance are independent quantities, and different two parameter statistical distribution functions are needed to describe these situations. The most celebrated such function is the Gaussian, or normal, distribution ... 

Using the logistic regression model, the deviation between the expectation of a particular binary observation (the conditional mean, E(Y x)) and the true value, 0 or 1, can be denoted by the random variable e, as shown in the following equation y = E(Y x) + e. Since y takes on only the values 0 and 1, e can take on only two possible values 1 - k x) when y = 1 and -n(x) when y = 0. Thus, the error term, e, follows a binomial distribution with mean 0 and variance n x) - Tt x)] (14). 

Given these assumptions, the distribution of the observed total number of counts according to probability theory should be binomial with parameters N and p. Because p is so small, this binomial distribution is approximated very well by the Poisson distribution with parameter Np, which has a mean of Np, and a standard deviation of Np. The mean and variance of a Poisson distribution are numerically equal so, a single counting measurement provides an estimate of the mean of the distribution Np and its square root is an estimate of the standard deviation /Np. When this Poisson approximation is valid, one may estimate the standard uncertainty of the counting measurement without repeating the measurement (a Type B evaluation of uncertainty). 

Figure 16.1 shows median width of a 95% Cl for risk difference (Figure 16.1a), risk ratio (Figure 16.1b), and odds ratio (Figure 16.1c). We can see that the width of Cl dramatically decreases as the number of studies increases regardless of the measure. Also, the plots of p = 0.4 are generally above those corresponding to = 0.2 and 0.3, particularly, for relative risk and odds ratio. This is expected since the variance of a binomial distribution is larger as the probability is closer to 0.5.

In words, at every time point r 0 the quantity of component X can be given by a binomial distribution with expectation JoCxp(-kt) and variance 7oexp( —A 0(1 exp( —A /)). As can be seen, the expectation agrees with the well-known deterministic solution. The two models are said to be consistent in the mean (Bartholomay, 1957). 

As regards the example mentioned in remark ( 16), the above property means that if there are two heaps of identical radionuclei, then not only the separate counts have binomial distribution (with expected values ip and nzp as well as variances nipq and 2p > respectively), but the total number of counts as well (with expected value n + n2)p and variance (mi + 02)pq). Choosing 1 s as the time of observation, the addition theorem expresses the additive property of activity. 

Normal approximation (De Moivre-Laplace limit theorem). It follows from the interpretation as well as from the central limit theorem that for large enough values of npq (npq > 6 suffices already) the binomial distribution can be approximated by a normal distribution with expected value p = p and variance [Pg.416]

In fact, the variability in rates may be obtained without recourse to numerical sampling. Focusing on the true positive rate, T 0), the distribution of true positive rates over the bootstrap samples is described by a binomial distribution, which is well approximated for even moderate amounts of data by a normal density with mean T d) and variance... 

The next step is to use the central limit theorem. Since the binomial distribution has a finite mean and variance, the adjusted sample average... 

The mean, variance, skewness, and kurtosis of the binomial distribution are given in the following equations ... 

As in the underlying binomial distribution, There is the mean value of x and is equal to np where n is the number of trials. The variance of a Poisson distribution also is np, as you can see by letting the factor (1 — p) in Eq. (5.78) go to 1. The standard deviation from the mean of a Poisson distribution (rr) thus is the square root of the mean. 

Poisson Distribution pwa- ls6"- [Sim n D. Poisson t 1840 French, mathematician] (1922) A probability density function that is often used as a mathematical model of the number of outcomes obtained in a suitable interval of time and space, that has its mean equal to its variance, that is used as a approximation to the binomial distribution, and that has the form... 

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