Independence
In an earlier chapter, we defined independence of two categorical variables — the conditional probabilities for one variable do not depend on the other and are also equal to its marginal probabilities.
We also mentioned independence in the context of random samples. When sampling from an infinite population or from a finite population with replacement, successive values are independent.
In general, X and Y are independent if knowing the value of X tells you nothing about the distribution of Y.
Mathematically, two variables X and Y are independent if the conditional distribution of Y given X = x does not depend on the value of x.
If X and Y are numerical variables, independence implies that their correlation coefficient will be zero.
Independence of mean and variance
Although we cannot prove it, the following result is important.
The sample mean and sample variance of random samples from a normal population are independent.
Simulation
The diagram below shows a random sample from a normal population. The sample mean and standard deviation are plotted on the scatterplot on the right.
Click Accumulate and take several samples. The scatterplot supports the fact that the sample mean and sample standard deviation are independent — there is a fairly circular scatter of crosses.
After 50 or more samples, the correlation coefficient between the mean and standard deviation should be close to zero.
Note that independence of the mean and standard deviation also implies independence of the mean and variance.