Lukacs's proportion-sum independence theorem
Okay kiddo, so let me explain the Lukacs's proportion-sum independence theorem to you in a way you can understand.
Do you know what a proportion is? It's when we compare one part of something to the whole thing. For example, if we have 5 red apples and 10 green apples, the proportion of red apples would be 5/15 (which simplifies to 1/3).
Now, do you know what independence means? It's when two things don't affect each other. For example, if you flip a coin and roll a die, the result of one doesn't depend on the result of the other.
So, Lukacs's proportion-sum independence theorem tells us that if we have a bunch of proportions (like the 5/15 example) and add them together, the result won't depend on each individual proportion.
Let's say we have two bowls of fruit. Bowl A has 3 bananas and 2 apples, and Bowl B has 2 bananas and 4 apples. The proportion of bananas in Bowl A is 3/5, and the proportion of bananas in Bowl B is 2/6 (which simplifies to 1/3). If we add these proportions together, we get 3/5 + 1/3 = 18/30 + 10/30 = 28/30 (which simplifies to 14/15).
Now, let's say we switch the fruits in Bowl A and Bowl B. Bowl A now has 2 bananas and 4 apples, and Bowl B has 3 bananas and 2 apples. The proportion of bananas in Bowl A is now 2/6 (which simplifies to 1/3), and the proportion of bananas in Bowl B is now 3/5. If we add these proportions together, we get 1/3 + 3/5 = 10/30 + 18/30 = 28/30 (which simplifies to 14/15) again!
So you see, even though we switched the fruits in the bowls, the result of adding the proportions of bananas didn't change. That's what Lukacs's proportion-sum independence theorem is all about - the result doesn't depend on the order or arrangement of the proportions.
Do you know what a proportion is? It's when we compare one part of something to the whole thing. For example, if we have 5 red apples and 10 green apples, the proportion of red apples would be 5/15 (which simplifies to 1/3).
Now, do you know what independence means? It's when two things don't affect each other. For example, if you flip a coin and roll a die, the result of one doesn't depend on the result of the other.
So, Lukacs's proportion-sum independence theorem tells us that if we have a bunch of proportions (like the 5/15 example) and add them together, the result won't depend on each individual proportion.
Let's say we have two bowls of fruit. Bowl A has 3 bananas and 2 apples, and Bowl B has 2 bananas and 4 apples. The proportion of bananas in Bowl A is 3/5, and the proportion of bananas in Bowl B is 2/6 (which simplifies to 1/3). If we add these proportions together, we get 3/5 + 1/3 = 18/30 + 10/30 = 28/30 (which simplifies to 14/15).
Now, let's say we switch the fruits in Bowl A and Bowl B. Bowl A now has 2 bananas and 4 apples, and Bowl B has 3 bananas and 2 apples. The proportion of bananas in Bowl A is now 2/6 (which simplifies to 1/3), and the proportion of bananas in Bowl B is now 3/5. If we add these proportions together, we get 1/3 + 3/5 = 10/30 + 18/30 = 28/30 (which simplifies to 14/15) again!
So you see, even though we switched the fruits in the bowls, the result of adding the proportions of bananas didn't change. That's what Lukacs's proportion-sum independence theorem is all about - the result doesn't depend on the order or arrangement of the proportions.