# The Strong Law of Large Numbers

• Nov 15th 2005, 07:29 PM
ThePerfectHacker
The Strong Law of Large Numbers
Can somebody in basic terms-for I have never studied probability- explain what the "Strong Law of Large Numbers" as well as "Weak Law of Small Numbers".
• Nov 17th 2005, 03:41 AM
hpe
Quote:

Originally Posted by ThePerfectHacker
Can somebody in basic terms-for I have never studied probability- explain what the "Strong Law of Large Numbers" as well as "Weak Law of Small Numbers".

The Law of Large Numbers says that the long-time sample mean of a series of independent observations approaches the theoretical mean of a single observation. That is, the sample mean ( a random quantity) becomes deterministic in the limit of large samples. It connects axiomatically based probability theory back to the frequentist definition of probability, an older approach.
• Nov 17th 2005, 08:37 AM
MathGuru
I remember in College statistics we learned never to have a sample larger than 10% of the poulation. Why is that? Does it have anything to do with the aforementioned theory?
• Nov 17th 2005, 09:53 AM
hpe
Quote:

Originally Posted by MathGuru
I remember in College statistics we learned never to have a sample larger than 10% of the poulation. Why is that? Does it have anything to do with the aforementioned theory?

Indirectly, yes. If your sample size is too large relative to the population, the members of the samples are "too dependent" (since you won't use the sample population member twice, your population changes as you are sampling). Then most of the usual theory becomes inapplicable. However, the LLN really only applied to cases where you can repeat a random experiment as many times as you want. So for a finite population, this works only if you are willing to sample the same member of the population more than once (sampling with replacement).
• Nov 18th 2005, 09:09 AM
ThePerfectHacker
I found it interesting that people used The Strong Law for millenia by observing patterns in nature without really understanding why it is thus.
Why is the Strong Law an important law in probabilty theory?
• Nov 18th 2005, 09:21 AM
hpe
Quote:

Originally Posted by ThePerfectHacker
I found it interesting that people used The Strong Law for millenia by observing patterns in nature without really understanding why it is thus.
Why is the Strong Law an important law in probabilty theory?

Because modern probability theory is axiomatic and not a priori concerned with reality. The LLN (any version, there are many) allows a connection back to reality.
The text is a series of forum posts discussing the Strong Law of Large Numbers and related concepts. It includes a question about the importance of the Strong Law in probability theory and an explanation of its significance.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# The Strong Law of Large Numbers

• Nov 15th 2005, 07:29 PM
ThePerfectHacker
The Strong Law of Large Numbers
Can somebody in basic terms-for I have never studied probability- explain what the "Strong Law of Large Numbers" as well as "Weak Law of Small Numbers".
• Nov 17th 2005, 03:41 AM
hpe
Quote:

Originally Posted by ThePerfectHacker
Can somebody in basic terms-for I have never studied probability- explain what the "Strong Law of Large Numbers" as well as "Weak Law of Small Numbers".

The Law of Large Numbers says that the long-time sample mean of a series of independent observations approaches the theoretical mean of a single observation. That is, the sample mean ( a random quantity) becomes deterministic in the limit of large samples. It connects axiomatically based probability theory back to the frequentist definition of probability, an older approach.
• Nov 17th 2005, 08:37 AM
MathGuru
I remember in College statistics we learned never to have a sample larger than 10% of the poulation. Why is that? Does it have anything to do with the aforementioned theory?
• Nov 17th 2005, 09:53 AM
hpe
Quote:

Originally Posted by MathGuru
I remember in College statistics we learned never to have a sample larger than 10% of the poulation. Why is that? Does it have anything to do with the aforementioned theory?

Indirectly, yes. If your sample size is too large relative to the population, the members of the samples are "too dependent" (since you won't use the sample population member twice, your population changes as you are sampling). Then most of the usual theory becomes inapplicable. However, the LLN really only applied to cases where you can repeat a random experiment as many times as you want. So for a finite population, this works only if you are willing to sample the same member of the population more than once (sampling with replacement).
• Nov 18th 2005, 09:09 AM
ThePerfectHacker
I found it interesting that people used The Strong Law for millenia by observing patterns in nature without really understanding why it is thus.
Why is the Strong Law an important law in probabilty theory?
• Nov 18th 2005, 09:21 AM
hpe
Quote:

Originally Posted by ThePerfectHacker
I found it interesting that people used The Strong Law for millenia by observing patterns in nature without really understanding why it is thus.
Why is the Strong Law an important law in probabilty theory?

Because modern probability theory is axiomatic and not a priori concerned with reality. The LLN (any version, there are many) allows a connection back to reality.
user
The text is a series of forum posts discussing the Strong Law of Large Numbers and related concepts. It includes a question about the importance of the Strong Law in probability theory and an explanation of its significance.

TYPE: SINGLE_QUESTION_WITH_ANSWER