Several theorems stating that sample mean converges to the expected value as $n\to\infty$. There is a weak law and a strong law of large numbers.
Questions tagged [law-of-large-numbers]
171 questions
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Law of large numbers: other formulae
Let $X_1, X_2, \ldots $ be an infinite sequence of i.i.d. random variables with $E(X_i)=\mu$ and $\mbox{Var}(X_i) < \infty$.
The law of large numbers states $\lim_{n \rightarrow \infty} \sum_{i=1}^{n} \frac{X_i}{n} = \mu$.
Wikipedia mentions that…
GCru
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Does coin tossing obey mean reversion?
If a coin comes up Heads 90 times out of the first 100 tosses, should one expect Tails to make a comeback over the next 100?
Reference from this page: https://www.financialwisdomforum.org/gummy-stuff/coin-tossing.htm
It is my understanding from the…
Ehsan
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Limit of the sum of independent identically distributed random variables
The question I'm working on says:
Let $X_1, X_2, \cdots$ be iid random variables each with mean $\mu$ and variance $\sigma^2$.
a) Determine
$$ \lim\limits_{n \to \infty} \frac{X_1^2 + \cdots + X_n^2}{n}$$
I think that the answer involves the strong…
russloewe
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Difference between uniform laws of large numbers and law of large numbers
How to understand the difference between uniform laws of large numbers and law of large numbers? In particular, what does the word "uniform" mean?
Ben
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Question about law of large numbers derivation
I am struggling with a small part of the proof of the law of large numbers.
I understand from Markov's inequality:
$$P(X\ge t) \le \frac{E(X)}{t} $$
and therefore if $ X = (\bar{Y} - E(Y))^2$
$$P(|\bar{Y} - E(Y)|\ge t) \le…
user3084100
- 525
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Law of Large Numbers under Transformation
The Law of Large Number, draw i.i.d examples of a random variable y, then with propability of 1 the average of y_1, ... y_n goes to the expected value of y.
When i apply a function to the y_i's, does then also hold that the average of f(y_1), ...…
user3680510
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Why does $\mathbb{E}(\frac{X_1+...+X_n}{n})=\mathbb{E}(X)$?
Let $X_1$, $X_2$, $...$, be i.i.d. and follow the same distribution as a random variable $X$ that has an expectation $\mathbb{E}(X)$ and a finite variance $\operatorname{Var}(X)$.
Why does it follow…
mavavilj
- 4,109