Do all distributions have the same mean?
Any normally distributed population will have the same proportion of its members between the mean and one standard deviation below the mean. Converting the values of the members of a normal population so that each is now expressed in terms of standard deviations from the mean makes the populations all the same.
When comparing two distributions it would be best to use relative frequency?
For the purpose of visually comparing the distribution of two data sets, it is better to use relative frequency rather than a frequency histogram since the same vertical scale is used for all relative frequency–from 0 to 1.
How to compare two distributions in real life?
The red line is the actual test statistic and the green line is the test statistic for 1000 random normal variables. By inserting the KS test statistic for the actual sample (i.e. the red line), we can see that the actual KS test statistic is contained inside the distribution.
How to compare a sample with a distribution?
When we compare a sample with a theoretical distribution, we can use a Monte Carlo simulation to create a test statistics distribution. For instance, if we want to test whether a p-value distribution is uniformly distributed (i.e. p-value uniformity test) or not, we can simulate uniform random variables and compute the KS test statistic.
How to calculate the density of two distributions?
As demonstrated here, convolving the two derives the density or mass of their sum, the random variable Z = X + Y. (Note that this only holds when X and Y are independent.) I suppose one could view f ( x) g ( x) as the Bayes factor of two candidate models, both with static parameters, after a single observation x.
How is the KS test used to compare two distributions?
As a non-parametric test, the KS test can be applied to compare any two distributions regardless of whether you assume normal or uniform. In practice, the KS test is extremely useful because it is efficient and effective at distinguishing a sample from another sample, or a theoretical distribution such as a normal or uniform distribution.