Nếu không gian là onedimensional (ví dụ độ dài của một sợi bông cùng những sai sót có thể xảy ra với xác suất không đổi ở tất cả các điểm), tương tự là ngay lập tức. Với không gian hai chiều (ví dụ một slide kính hiển vi mà vi khuẩn được phân phối ngẫu nhiên với kỹ thuật pha trộn hoàn hảo) | The Poisson distribution 73 situation in which particles are randomly distributed in space. If the space is onedimensional for instance the length of a cotton thread along which flaws may occur with constant probability at all points the analogy is immediate. With two-dimensional space for instance a microscopic slide over which bacteria are distributed at random with perfect mixing technique the total area of size A may be divided into a large number n of subdivisions each of area A n the argument then carries through with A replacing T. Similarly with three-dimensional space bacteria well mixed in a fluid suspension the total volume V is divided into n small volumes of size V n. In all these situations the model envisages particles distributed at random with density X per unit length area or volume . The number of particles found in a length area or volume of size I A or Ẹ will follow the Poisson distribution where the parameter p X XA or XF . The shapes of the distribution for p 1 4 and 15 are shown in Fig. . Note that for p 1 the distribution is very skew for p 4 the skewness is much less and for p 15 it is almost absent. The distribution is determined entirely by the one parameter p. It follows that all the features of the distribution in which one might be interested are functions only of p. In particular the mean and variance must be functions of p. The mean is E x E xF V this result following after a little algebraic manipulation. By similar manipulation we find E x2 p2 p and var x E x2 - p2 19 p Thus the variance of X like the mean is equal to p. The standard deviation is therefore ựp. Much use is made of the Poisson distribution in bacteriology. To estimate the density of live organisms in a suspension the bacteriologist may dilute the suspension by a factor of say 10 5. take samples of say 1 cm3 in a pipette and drop the contents of the pipette on to a plate containing a nutrient medium on which the bacteria grow. After some time each .
