Cayley theorem proof @DrJuanKlopper

Uploaded October 2018 | Updated September 2026, 2 weeks ago

In this video I show you how to prove Cayley's theorem, which states that every group is isomorphic to a permutation group.

This video is a bit long because I take the time to revisit all the concepts required in the proof. these include isomorphisms, injective, surjective, and bijective functions, and how to construct the set of elements in the given group to construct the required permutation group.

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$Bootstrap resampling for confidence intervals and sampling distrubutions #DataScience #Bootstrap #Resampling #Julia #VSCode #Confidence #Intervals #Probability In this video I show you how to think about using bootstrap resampling to express uncertainty in a statistic and also how to think about the sampling distribution of a test statistic and the probability of that test statistic. Using the Julia Language in Visual Studio Code we generate values for a continuous numerical variable for two groups (generated from the two sample space elements of a categorical variable). Our test statistic is the difference in means for the variable between the two groups. We generate a sampling distribution of the test statistic under the assumption of the null hypothesis. The latter states that there is no difference between the two means, that is, the test statistic is zero. Given this assumption we can randomly reassign subjects to the two groups and recalculate the test statistic. Doing this many times over produces the sampling distribution. We can then see how many of the resampled tests statistics are below the negative of our test statistic and how many are more than positive our test statistic. Adding these and expressing it as a fraction of the number of resamples gives us the p value for our statistic. We also express the mean for the variable in both groups as a whole. We must consider the uncertainty in this statistic. We estimate the lower and upper bounds for the parameter in the population. This is the confidence interval.$

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