&gt; I&#x27;m going to have to look into what you mean by using intuition to judge independence. Any good sources on that?<p>If look at the relevant theorems, then will conclude
that intuitively random variables X and Y are independent if and only if knowledge of one of them does nothing to tell you more about the other one.<p>If don&#x27;t have independence, then can often have conditional independence, e.g., random variable X, Y are conditionally independent given Z implies<p>E[XY|Z] = E[X|Z] E[Y|Z]<p>So,<p>X = number of auto accidents today in LA<p>Y = number of auto accidents today in NY.<p>----------<p>In a family with two children<p>X = gender of the first child<p>Y = gender of the second child<p>---------<p>At a Web site<p>X = number of new user sessions from 10:00 AM to 10:01 AM<p>Y = number of new user sessions from 10:01 AM to 10:02 AM<p>-----------<p>On a commercial airline flight<p>X = height of the first passenger to buy a ticket<p>Y = height of the 10th passenger
to buy a ticket<p>Etc.  Just intuition from knowledge of one of these two tells you nothing more about the other one (we assume we know the two distributions -- that we get insight into a distribution doesn&#x27;t count).\<p>Then using independence and very little more, can use the main limit theorems, law of large numbers, central limit theorem, etc.<p>If don&#x27;t have independence, then sometimes can fake it:  Given a list of n phone numbers, if toss them into a bucket and stir it and pull out the numbers and label them as X_1, X_2, ..., then in common situations can assume that the X&#x27;s are independent.<p>There are tests for independence.  Likely the first to know about is a chi-squared test on a table of values of the two variables.