Statistics and Optimization

Author: Chief Editor: Qin Mingda et al
Publisher:
Publish Date: 2004-01-01
Features:
Fragment: 2. Histogram We further use graphics to intuitively reflect the frequency (relative frequency) distribution of the sample. On the xy-plane, draw a row of vertical rectangles: for each i (i = 0, 1, …, m), take the subinterval (ti, ti + 1] as the base and yi = fi/l as the height. For example, in Example 1, yi = fi/l = ni / (80 × 0.05) (i = 0, 1, …, 8), calculate the heights of 9 rectangles to form a histogram, which is a simplified term for a histogram. It can roughly describe the probability distribution of the population X because the area of each vertical rectangle is Ai = fi/l · l = fi (i = 0, 1, …, m). According to the Bernoulli's Law of Large Numbers in probability theory, when the sample size n is sufficiently large, the frequency fi approximates the probability that the random variable X (population) takes a value in the interval (ti, ti + 1], i.e., f(x) is the probability density function of X. With the histogram, we can further draw the approximate shape of the probability density curve of the population X. We use the "gain and compensate" method to sketch a smooth curve, i.e., when drawing the curve, we try to ensure that the area of each rectangle outside the curve equals the area inside the curve, while maintaining the smoothness of the curve, as shown in Figure 1-1. The curve obtained in this way is an approximate graphical representation of the probability density function f(x) of X. The curve in Figure 1-1 resembles the density curve of a normal distribution, reflecting that the height of seedlings X asymptotically follows a normal distribution. Intuitively, we can consider the growth of seedlings to be normal. However, this is only an intuitive judgment, and there are strict testing methods later. It is easy to see that if the sample size is larger (i.e., n is larger) and the grouping is finer (i.e., m is larger), the histogram will be closer to the "curved trapezoid" under the probability density curve, and the smooth curve drawn will be closer to the probability density curve. The histogram method is only applicable to the case of continuous populations. For a discrete population X, we can also create an approximate graphical representation of the distribution law of X. First, list the distinct possible values of the sample values x1, x2, …, xn in ascending order as t1 < t2 < … < tk (k ≤ n), and use the "tallying" method to count the frequencies of x1, x2, …, xn taking values at each tj (j = 1, 2, …, k). Organize the results into a form similar to Table 1-2 (replace the subinterval (ti, ti + 1] with the discrete value ti), and then draw a frequency distribution graph similar to a histogram, which is an approximate graphical representation of the distribution law of the population X (see Figure 1-2).
3. Empirical Distribution Function Here, we introduce an empirical distribution function that is applicable to both discrete and continuous populations. It is a good approximation of the distribution function of the population X.