Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [law-of-large-numbers]

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.

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Why law of large numbers does not apply in the case of Apple share price?

Here is the article in NY times called "Apple confronts the law of large numbers". It tries to explain Apple share price rise using law of large numbers. What statistical (or mathematical) errors does this article make?
mpiktas
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Why should the frequency of heads in a coin toss converge to anything at all?

Suppose we have any kind of coin. Why should the relative frequency of getting a heads converge to any value at all? One answer is that this is simply what we empirically observe this to be the case, and I think this is a valid answer. However, my…
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Expectation of 500 coin flips after 500 realizations

I was hoping someone could provide clarity surrounding the following scenario. You are asked "What is the expected number of observed heads and tails if you flip a fair coin 1000 times". Knowing that coin flips are i.i.d. events, and relying on the…
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What theories should every statistician know?

I'm thinking of this from a very basic, minimal requirements perspective. What are the key theories an industry (not academic) statistician should know, understand and utilize on a regular basis? A big one that comes to mind is Law of large…
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Central limit theorem and the law of large numbers

I have a very beginner's question regarding the Central Limit Theorem (CLT): I am aware that the CLT states that a mean of i.i.d. random variables is approximately normal distributed (for $n \to \infty$, where $n$ is the index of the summands) or…
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Central limit theorem versus law of large numbers

The central limit theorem states that the mean of i.i.d. variables, as $N$ goes to infinity, becomes normally distributed. This raises two questions: Can we deduce from this the law of large numbers? If the law of large numbers says that the mean…
user9097
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Law of Large Numbers for whole distributions

I'm aware of the Law(s) of Large Numbers, concerning the means. However, intuitively, I'd expect not just the mean, but also the observed relative frequencies (or the histogram, if we have a continuous distribution) to approach the theoretical…
Igor F.
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Is there a law that says if you do enough trials, rare things happen?

I'm trying to make a video about loaded dice, and at one point in the video we roll about 200 dice, take all the sixes, roll those again, and take all the sixes and roll those a third time. We had one die that came up 6 three times in a row, which…
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When does the law of large numbers fail?

The question is simply what is stated in the title: When does the law of large numbers fail? What I mean is, in what cases will the frequency of an event not tend to the theoretical probability?
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What is the difference between the Monte Carlo (MC) and Monte Carlo Markov Chain (MCMC) method?

The goal of both methods seems to be to derive an estimate of a posterior/target distribution. If a process model exists which links some input parameters (which are themselves uncertain and can be described by a PDF) to an output parameter through…
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Weak law of large numbers - redundant?

I might be missing something basic - but it appears that the strong law of large numbers covers the weak law. If that case, why is the weak law needed?
user18496
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Difference between the Law of Large Numbers and the Central Limit Theorem in layman's term?

Almost all books teach Law of Large Numbers first then the Central Limit Theorem one next. But what are the relationship and differences between two theorem? My attempt: Here is my understanding (Informally), Law of Large Numbers says if we have…
Haitao Du
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Is sample mean the "best" estimation of distribution mean in some sense?

By (weak/strong) law of large numbers, given some iid sample points $\{x_i \in \mathbb{R}^n, i=1,\ldots,N\}$ of a distribution, their sample mean $f^*(\{x_i, i=1,\ldots,N\}):=\frac{1}{N} \sum_{i=1}^N x_i $ converges to the distribution mean both in…
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Intuition behind strong vs weak laws of large numbers (with an R simulation)

This rather looks quite basic, but when referring to weak and strong law of large numbers this is the definition I look at (Casella and Berger) Can you please give an 'intuition' in understanding the difference between them. Also, what does the…
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WLLN: can expectation exist but be infinite?

Weak law of large numbers: Let $\{h_i, i = 1, \dots n\}$ be an $m \times q$ sequence of iid random variables with mean $\mu = E[h_i]$ that exists and is finite. Then $1/n \sum_{i = 1}^n h_i \rightarrow \mu$ in probability. I don't understand why we…
Kolibris
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