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Continuous Distributions
Published in Michael Baron, Probability and Statistics for Computer Scientists, 2019
This is resolved by introducing a continuity correction. Expand the interval by 0.5 units in each direction, then use the Normal approximation. Notice that PX(x)=P{X=x}=P{x−0.5<X<x+0.5} is true for a Binomial variable X; therefore, the continuity correction does not change the event and preserves its probability. It makes a difference for the Normal distribution, so every time when we approximate some discrete distribution with some continuous distribution, we should be using a continuity correction. Now it is the probability of an interval instead of one number, and it is not zero.
Probability Functions
Published in Edward G. Schilling, Dean V. Neubauer, Acceptance Sampling in Quality Control, 2017
Edward G. Schilling, Dean V. Neubauer
When the product np is greater than 5, the binomial distribution may be approximated by the normal distribution with μ=np,σ=npq Note that nq must also be greater than 5. However, in acceptance sampling, it is usually the case that q > p. Here, the normal cumulative probability is taken over a region corresponding to the number of successes desired. In approximating a discrete distribution, such as the binomial, with a continuous distribution, such as the normal, it is necessary to use a “continuity” correction. Since the probability of a point in a continuous distribution is zero, it is necessary to approximate each discrete number of successes by a band on the x-axis going out from the number one-half units on each side as shown in Figure 3.10. Thus, the probability of x or less successes would be found as the area up to x + 1/2 under the normal curve. The probability of x or more successes would be the area above x −1/2 and so on.
Computing Probabilties for Rank Statistics Used with Block Design Nonparametric Subset Selection Rules
Published in American Journal of Mathematical and Management Sciences, 2022
How well does the normal asymptotic approximation, given in (7) and (8), fit the exact values of the cdf G(d;k,n)? The applications of the nonparametric block design selection problems (e.g., McDonald, 2016) utilize this approximation to obtain the appropriate percentile for the test statistic. To answer the question, the asymptotic approximation is compared to the simulation approximation summarized in the Tables 4 and 5. The R code given in Appendix B facilitates this comparison. This code consists of two parts. The first part simulates G(d;k,n) as described earlier in this Section. The second part computes the cdf using the asymptotic normality. The integral in (7), modified by increasing d by 0.5, is evaluated by stochastic integration. A large number (specified by msim) of standard normal random variables (mean = 0 and standard deviation = 1) are drawn and denoted by x. For each of these, the quantity is calculated. Then the value of the expectation, where X is a standard normal random variable, is approximated by the average of the msim values For these calculations, the value of d in (7) has been been replaced by (d + 0.5) for inclusion of a continuity correction factor (CCF), similar to what is done in approximating binomial cdf probabilities with a normal approximation. In the case here, the inclusion of the CCF has improved slightly the approximations.