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Boole inequality and calculation of union probability of n arbitrary events. Explain in a simple way the concept of sampling distribution of the mean (or any other computable statistics on the sample as standard deviation (sigma) or mode or median).

Boole inequality and calculation of union probability of n arbitrary events

The Boole inequality, or union bound, says that for every limited or countable collection of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events. Formally, if we have a finite or countable set of events A1, A2, A3, An we say that:

Bool inequality

It is easily demonstrable for n = 2 event. If for example, we have two arbitrary events A and B we can say for the first axiom of probability that:

P(AUB)= P(A)+P(B)- P(A∩B) 

P(A∩B) is >0, which is subtracted, and so it follows that P(AUB)<= P(A)+P(B).

If we consider the event C = AUB and another arbitrary event D we can iterate the demonstration for n = n+1. Therefore the result is the above formula. 

So it is possible to apply the Boole inequality to compute the union probability of n arbitrary event. Here’s the formula:

Calculation of union probability of n arbitrary events
Explain in a simple way the concept of the sampling distribution of the mean (or any other computable statistics on the sample as standard deviation (sigma) or mode or median).

The mean of the sampling distribution of the mean is the mean of the population from which the scores were sampled. Therefore, if a population has a mean μ (unknown), then the mean of the sampling distribution of the mean is also μ. The symbol μM (calculated) is used to refer to the mean of the sampling distribution of the mean. Therefore, the formula for the mean of the sampling distribution of the mean can be written as:   μM = μ

It is important to keep in mind that every statistic, not just the mean, has a sampling distribution.

 

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