1. 3.3 Important Continuous Distributions
1.1. 3.3.1 The Normal Distribution
1.2. 3.3.2 The Lognormal Distribution
1.2.1. Variables in a system sometimes follow an exponential relationship, say x = exp(w). If the exponent is a random variable, say w, x = exp(w) is a random variable and the distribution of x is of interest. An important special case occurs when w has a normal distribution. In that case, the distribution of x is called a longnormal distribution.
1.3. 3.3.3 The Exponential Distribution
1.3.1. The exponential distribution is widely used in the field of reliability engineering as a model of the time to failure of a component or system. In these applications, the parameter is called the failure rate of the system, and the mean of the distribution is called the mean time to failure
1.4. 3.3.4 The Gamma Distribution
1.5. 3.3.5 The Weibull Distribution
1.5.1. The Weibull distribution has been used extensively in reliability engineering as a model of time to failure for electrical and mechanical components and systems
2. 3.4 Probability Plots
2.1. 3.4.1 Normal Probability Plots
2.1.1. Probability plotting is a graphical method for determining whether sample data conform to a hypothesized distribution based on a subjective visual examination of the data
2.1.2. Probability plotting typically uses special graph paper, known as probability paper, that has been designed for the hypothesized distribution. Probability paper is widely available for the normal, longnormal, Weibull, and various chi-square and gamma distributions
2.1.3. A very important application of normal probability plotting is in verification of assumptions when using statistical inference procedures that require the normality assumption. This will be illustrated subsequently.
2.2. 3.4.2 Other Probability Plots
2.2.1. Probability plots are extremely useful and are often the first technique used when we need to determine which probability distribution is likely to provide a reasonable model for data
3. 3.5 Some Useful Approximations
3.1. 3.5.1 The Binomial Approximation to the Hypergeometric
3.1.1. This approximation is useful in the design of acceptance-sampling plans. Recall that the hypergeometric distribution is the appropriate model for the number of nonconforming items obtained in a random sample of n items from a lot of finite size N. Thus, if the sample size n is small relative to the lot size N, the binomial approximation may be employed, which usually simplifies the calculations considerably.
3.2. 3.5.2 The Poisson Approximation to the Binomial
3.2.1. That the Poisson distribution could be obtained as a limiting form of the binomial distribution for the case where p approaches zero and n approaches infinity with Lamda constant
3.3. 3.5.3 The Normal Approximation to the Binomial
3.3.1. we defined the binomial distribution as the sum of a sequence of n Bernoulli trials, each with probability of success p. If the number of trials n is large, then we may use the central limit theorem to justify the normal distribution with mean np and variance np(1 − p) as an approximation to the binomial.
3.4. 3.5.4 Comments on Approximations
3.4.1. In this figure, H, B, P, and N represent the hypergeometric, binomial, Poisson, and normal distributions, respectively. The widespread availability of modern microcomputers, good statistics software packages, and hand-held calculators has made reliance on these approximations largely unnecessary, but there are situations in which they are useful, particularly in the application of the popular three-sigma limit control charts.
4. 3.1 Describing Variation
4.1. 3.1.1 The Steam-and-Leaf Plot
4.1.1. No two units of product produced by a process are identical. Some variation is inevitable.
4.1.2. One of the most useful graphical techniques is the stem-and leaf display.
4.2. 3.1.2 The Histogram
4.2.1. To construct a histogram for continuous data, we must divide the range of the data into intervals, which are usually called : Class Intervals, Cells, Bins.
4.2.2. Bins should be equal width to enhance the visual information in the histogram.
4.2.3. The histogram as a technique best suited for larger data sets, containing 75 to 100 or more observations.
4.3. 3.1.3 Numerical Summary of Data
4.3.1. The most important measure of central tendency in the sample is the sample average
4.3.2. The variability in the sample data is measured by the sample variance
4.3.3. The units of the sample variance s^2 are the square of the original units of the data. This is often inconvenient and awkward to interpret, and so we usually prefer to use the square root of s^2, called the sample standard deviation s, as a measure of variability s^2 = 180 2928 . A^2 s = 13 43 . A The standard deviation does not reflect the magnitude of the sample data, only the scatter about the average.
4.4. 3.1.4 The Box Plot
4.4.1. The box plot is a graphical display that simultaneously displays several important features of the data, such as location or central tendency, spread or variability, departure from symmetry, and identification of observations that lie unusually far from the bulk of the data (these observations are often called “outliers”)
4.5. 3.1.5 Probability Distribution
4.5.1. A probability distribution is a mathematical model that relates the value of the variable with the probability of occurrence of that value in the population. In other words, we might visualize layer thickness as a random variable, because it takes on different values in the population according to some random mechanism, and then the probability distribution of layer thickness describes the probability of occurrence of any value of layer thickness in the population
4.5.2. The standard deviation is a measure of spread or scatter in the population expressed in the original units. Two distributions with the same mean but different standard deviations are shown in Fig. 3.13.
5. 3.2 Important Discrete Distributions
5.1. 3.2.1 The Hypergeometric Distribution
5.1.1. The hypergeometric distribution is the appropriate probability model for selecting a random sample of n items without replacement from a lot of N items of which D are nonconforming or defective
5.2. 3.2.2 The Binomial Distribution
5.2.1. The binomial distribution is used frequently in quality engineering. It is the appropriate probability model for sampling from an infinitely large population, where p represents the fraction of defective or nonconforming items in the population
5.3. 3.2.3 The Poisson Distribution
5.4. 3.2.4 The Pascal and Related Distributions
5.4.1. The Pascal distribution, like the binomial distribution, has its basis in Bernoulli trials. Consider a sequence of independent trials, each with probability of success p, and let x denote the trial on which the r th success occurs.
5.4.2. The first of these occurs when and is not necessarily an integer. The resulting distribution is called the negative binomial distribution.
5.4.3. The other special case of the Pascal distribution is if r = 1, in which case we have the geometric distribution.