My Stats Journal

When a distribution is “Mixed”…

What are weights, and when to use them?

I will keep this short now.

Say, you have a distribution which tells you that a random Variable X — has probability 0.2 of being 0, 0.5 of being Y and 0.3 of being Z. (Where Y and Z are independent distributions — think of them like mutually exclusive alternatives to each other)

Then we can formulate the distribution by thinking about them as a mixture, with weights 0.2, 0.5 and 0.3 respectively of being 0, Y and Z respectively.

NOTE: The weights should always sum up to 1.

📌Example — There are three kinds of people

Say, in an institution there are three kinds of people — the good, the bad and the ugly. Their proportions are given like percentages or their numbers are given or some combination is given. (Anything that can be used to determine the probability of selecting a person randomly and him/her turning out to be one of those categories).

Now, you are told that they have some different policy amounts or ages or anything that is related to their attributes (good/bad/ugly) — having certain distributions.

Then we can say that the policy/age/other value is a random variable with the weights of the proportions of people. And their mean and variance can be calculated as well.

You may be told to narrow your analysis to only two of the classes (eg. Good and bad only) then you need to reconfigure the weights, such that the probability of selecting either good or bad is 1(hence, no other type exist in the view).

📌Example — It either happens or it doesn’t

A probability of something happening is, let’s say 0.3 and something not happening is 0.7.

Under the premise that it happened (given that it happened), we are told that it brings happiness (if it could be measured) whose amount follows a distribution of X.

Hence, we are asked to find the expected happiness. How silly right?

Well, you have to think of the weights now as it happens and it doesn’t happen.

This is so because happiness is X only when there is happiness, and X actually doesn’t contain the information about the happiness not being present (with 0.3 probability). Surely F(x) gives probabilities 0 to 1. But in the context of the entire question it only gives the probabilities linearly interpolated to 0 to 0.7.

Hence, if we wish to find the true probability happiness being some certain value we multiply 0.7(its weight) to the probability of X (of that certain value).