统计学 正态分布 总结

������ ������ ≥ ������ = ������ ������ ≥ no matter what the values of µ and σ.
As before, we can also find a value of X if we are given a probability for X.
A Summary of the Normal Distribution The normal distribution is a bell-shaped distribution which is defined by its probability density function, f(x). There is, theoretically, an infinite number of f(x)’s and, thus, an infinite number of different normal distributions, one normal distribution for each combination of values for the population mean, µ, and population standard deviation, σ. The normal distribution for the random variable with µ = 0 and σ = 1, is called the standard normal random variable, symbolized by Z. It is this random variable for which there is a table which allows us to find P(0 ≤ Z ≤ z) where z represents some value of Z. Knowing this probability and knowing that the normal distribution is a symmetrical distribution allows us to calculate any other probabilities concerning Z. By drawing the Z-distribution and by shading in the area representing the probability that we are looking for, we know that we can find these probabilities by: o reading the probability directly from the table (e.g., P(0 ≤ Z ≤ 1.35)) o adding .5 to the probability read from the table (e.g., P(Z ≥ -1.35)) o subtractinwk.baidu.com the probability read from the table from .5 (e.g., P(Z ≥ 1.35)) o summing two probabilities read from the table (e.g., P(-1.05 ≤ Z ≤ 1.35)) o subtracting one probability read from the table from another probability read from the table (e.g., P(1.05 ≤ Z ≤ 1.35)) We can also use the Z-table to find values of z given probabilities (e.g., P(Z ≥ z) = .0250) Because the mean of Z is 0 and its standard deviation is 1, a value of Z = z also tells us that this value of Z is z standard deviations from its mean (e.g., Z = 2 tells us that Z is 2 standard deviations from its mean). If X is normal with any mean, µ, and any standard deviation, σ, a value of X = x can be transformed to a value of Z = z, the number of standard deviations this value is from its mean, using the transformation: ������ = ������ − ������ ������ − ������ → ������ = ������ ������


And, it so happens that, as long as any normal random variable is z standard deviations from its mean, the probability that we are looking for is the same as long as z is the same, or, e.g., ������ − ������ ������