The law of large numbers is a very popular theorem that perfectly describes the result of experiments that are repeated a large number of times.
In statistics and probability theory, this law is regarded as one of the most dependent and reliable theorems if you need to study experiments that are repeated.
What is the Law of Large Numbers?
The law of large numbers states that if any experiment or study is being repeated a large number of times independently, the average of the results of the trial must come close to the value expected in the study.
The result of the repeated experiments comes closer to the expected value as the number of repetitions or trials is increased. This law is even applicable to random events. Thus, it becomes an important concept in the study of statistics since it states that the result of random events must also come closer to the expected stable long-term result.
One must make sure the theorem is used only on experiments that have a large number of trials. Also, the average of the results that have a small number of repetitions will not be close to the expected value.
For example, let’s take a dice. Here is the expected value of the dice events:
EV = (1+2+3+4+5+6)/6 = 3.5
Now the average result of rolling a dice for a large number of times will be closer to 3.5.
The law of large numbers is actively used in the finance industry. It is used to figure out the growth rates of businesses. When it comes to business growth, the law of large numbers state that the growth rate of a company declines as it continues to expand.
What is the Weak Law of Large Numbers?
The weak law of large numbers states that there is a high probability for the average of a large number of repetitions to be close to expected value, even when the specified margin is nonzero.
The weak law of large numbers is extensively used in statistics and probability as well. The law is used to perfectly describe how any sequence of probabilities from any experiment repeated a large number of times converges.
What is the Strong Law of Large Numbers?
While the weak law of large numbers is pretty much easy to prove, the strong law of large numbers is a little complex and is only present in advanced graduate texts. The strong law of large numbers is used to perfectly validate the relative frequency distribution of probability.
However, it doesn’t state anything when it comes to addressing the limiting distribution of the sum of the results of an experiment repeated a large number of times. The strong law of large numbers shows how the sequence of probabilities from any experiment repeated a large number of times behaves in the limit.
The law of large numbers is quite an intriguing topic in the study of statistics and probabilities and has found its application in many fields. We hope this piece of content helped you understand the law of large numbers to perfection. Let us know what you think about the theorem in the comments section. We’d love to hear from you!