# What is Central limit theorem: Definition and 52 Discussions

In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a bell curve) even if the original variables themselves are not normally distributed. The theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions. This theorem has seen many changes during the formal development of probability theory. Previous versions of the theorem date back to 1811, but in its modern general form, this fundamental result in probability theory was precisely stated as late as 1920, thereby serving as a bridge between classical and modern probability theory.
If

X

1

,

X

2

,

,

X

n

{\textstyle X_{1},X_{2},\dots ,X_{n}}
are

n

{\textstyle n}
random samples drawn from a population with overall mean

μ

{\textstyle \mu }
and finite variance

σ

2

{\textstyle \sigma ^{2}}
, and if

X
¯

n

{\textstyle {\bar {X}}_{n}}
is the sample mean, then the limiting form of the distribution,

Z
=

lim

n

n

(

X
¯

n

μ

σ

)

{\textstyle Z=\lim _{n\to \infty }{\sqrt {n}}{\left({\frac {{\bar {X}}_{n}-\mu }{\sigma }}\right)}}
, is a standard normal distribution.For example, suppose that a sample is obtained containing many observations, each observation being randomly generated in a way that does not depend on the values of the other observations, and that the arithmetic mean of the observed values is computed. If this procedure is performed many times, the central limit theorem says that the probability distribution of the average will closely approximate a normal distribution. A simple example of this is that if one flips a coin many times, the probability of getting a given number of heads will approach a normal distribution, with the mean equal to half the total number of flips. At the limit of an infinite number of flips, it will equal a normal distribution.
The central limit theorem has several variants. In its common form, the random variables must be identically distributed. In variants, convergence of the mean to the normal distribution also occurs for non-identical distributions or for non-independent observations, if they comply with certain conditions.
The earliest version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem.

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1. ### Limit of probabilities of a large sample

My first thought as well but I think the problem is deeper than that. I think that as the n tends towards infinity the probability of the the sample mean converging to the population mean is 1. Looking at proving this. By the Central Limit Theorem the sample mean distribution can be approximated...
2. ### I Why doesn't sinc(x) converge to Gaussian upon repeated convolution?

Hello, I've read that repeated convolution tends, under certain conditions, to Gaussian distribution. I found this description helpful, and Wikipedia's version of this says: The central limit theorem states that if x is in L1 and L2 with mean zero and variance ##σ^2##, then...
3. ### The t-test and the central limit theorem

Ans is 3. I know basic t-test but I have no clue to solve this question. Thanks.
4. ### I Central Limit Theorem and fitting data

I am working with a simulation which generates an arbitrary number ##n## of identical curves with different phases and calculates their (normalized) sum. As expected, the fluctuation depth of the curves decreases as we increase ##n##. Here is an example of my simulation (when ##n>1##, the...
5. ### I Central limit theorem, panel study

I am doing a panel study with multiple linear regression. When I want to make sure that the residuals are normally distributed, as is a requirement for the regression model, can I assume so due the Central limit theorem (given the size is sufficient)? Or does it not apply when there is a time...
6. ### MHB Probability & Central Limit Theorem

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual. μx̄ = μ = 12,749 σ = 1.2 n = 35 For the given sample n = 35, the probability of a sample mean being less than 12,749 or greater...
7. ### MHB Estimating the Probability of Earning a Certain Amount in a Weekend as a Waiter

A waiter believes the distribution of his tips has a model that is slightly skewed to the left, with a mean of $\$8.20$and a standard deviation of ​$\$5.60$. He usually waits on about 60 parties over a weekend of work. ​a) Estimate the probability that he will earn at least ​$\$600\$. ​b) How...
8. ### I think this is about the Central Limit Theorem

Homework Statement An engineer is measuring a quantity q. It is assumed that there is a random error in each measurement, so the engineer will take n measurements and reports the average of the measurements as the estimated value of q. Specifically, if Yi is the value that is obtained in the...
9. ### Central limit theorem and estimates of probability

Homework Statement Assume five hundred people are given one question to answer - the question can be answered with a yes or no. Let p =the fraction of the population that answers yes. Give an estimate for the probability that the percent of yes answers in the five hundred person sample is...
10. ### Philosophical question about central limit theorem

Well, this is probably a stupid question, but I don't see why (yet). Let Xi be random variables identically distributed, with mean 0, such that the cumulative distribution is = 0 for all -1 < x < 1. So, I believe it is clear that for all n, the cumulative distribution of Z = (X1 + X2 ... Xn)/n...
11. ### Central limit theorem - finding cumulants

Homework Statement Given: y=\frac{\sum_{i}x_i-N\left \langle x \right \rangle}{\sqrt{N}} Show that the cumulants of y are: \begin{matrix} \left \langle y \right \rangle_c=0& & \left \langle y^2 \right \rangle_c=\left \langle x^2 \right \rangle_c & & \left \langle y^m \right...
12. ### Central Limit Theorem applied to a Bernoulli distribution

As I understand it, one result of the central limit theorem is that the sampling distribution of means drawn from any population will be approximately normal. Assume the population consist of Bernoulli trials with a given probability p and we want to estimate p. Then our population consist of...
13. ### Central Limit Theorem Question

Homework Statement The Rockwell hardness of certain metal pins is known to have a mean of 50 and a standard deviation of 1.5. a)if the distribution of all such pin hardness measurements is known to be normal, what is the probability that the average hardness for a random sample of 9 pins is at...
14. ### Binomial Central Limit Theorem

Homework Statement Here are the problems: A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a speciﬁed number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the...
15. ### MHB Applying the central limit theorem

Suppose the time in days until a component fails has the gamma distribution with alpha = 5, and theta = 1/10. When a component fails, it is immediately replaced by a new component. Use the central limit theorem to estimate the probability that 40 components will together be sufficient to last at...
16. ### MHB Central Limit Theorem & Gamma Distribution

The time it takes to complete a project is a random variable Y with the exponential distribution with parameter β=2 hours. Apply the central limit theorem to obtain an approximation for the probability that the average project completion time of a sample of n=64 projects undertaken...
17. ### Find probability density function from Central Limit theorem

Homework Statement How can I derive the probability density function by using the Central Limit theorem? For an example, let's say that we have a random variable Xi corresponding to the base at the ith position; to make even simpler, let's say all probabilities are equal. If we have four...
18. ### Central Limit Theorem Proof: Expanding the Exponential

Hello. This is the most closely matching forum I found for this, so I hope my question fits here. I was looking at the following proof of the Central-Limit theorem: http://physics.ucsc.edu/~peter/250/deriv_climit.pdf and after Eq. (10) it says: "Expanding out the exponential in the last...
19. ### Central limit theorem [probability]

1. In any one-minute interval, the number of requests for a popular Web page is a Poisson random variable with expected value 360 requests. A Web server has a capacity of C requests per minute. If the number of requests in a one-minute interval is greater than C the server is overloaded. Use...
20. ### Using Chebyshev and other inequality formulas (maybe even Central Limit Theorem)

34. Turner's syndrome is a rare chromosomal disorder in which girls have only one X chromosome. It affects about 1 in 2000 girls in the United States. About 1 in 10 girls with Turner's syndrome also suffer from an abnormal narrowing of the aorta. a. In a group of 4000 girls, what is the...
21. ### Proof of central limit theorem

Hi I want to prove this using momentgenerating functions. I would like to do this without going into the standard normal distribution, just the normal distribution. I would like to show that the momentgenerating function of (x1+x2+x3...xn)/n--->e^(ut+sigma^2t/2) as n-->infinity. x1, x2...
22. ### Motivating the Central Limit Theorem

Hi, All: I will be teaching intro Stats next semester, and I always have trouble making the CLT seem relevant/meaningful to students without much math nor probability background, whose eyes glaze at the mention of the distribution of the sampling mean being normal, no matter (given random...
23. ### How to calculate how big of a sample size is needed for the Central Limit Theorem?

Is there a way to calculate/estimate how big a sample from a parent distribution would need to be for the distribution of the mean of that sample to be approximately normally distributed?
24. ### Central Limit Theorem and STEM

I keep reading explanations that say things like "the mean is normally approximated" but I don't know what that means. Are they saying that if you take a load of samples then plot the means of every one of those samples on a graph that the mean of that graph will be approximately the population...
25. ### What is the connection between X_n and Y_n?

Homework Statement [PLAIN]http://img263.imageshack.us/img263/8679/statsji.jpg The Attempt at a Solution I've done part (a) and I know what the CLT says but how does part (a) link with part (b) as if X_n \sim Bern(p) then \displaystyle \sum^n_{i=1} X_i \sim Bin(n,p) so X_n =...
26. ### Finding Probability of Low Cereal Content in Random Selection of Boxes

Homework Statement A machine fills cereal boxes at a factory. Due to an accumulation of small errors (different flakes sizes, etc.) it is thought that the amount of cereal in a box is normally distributed with mean 22 oz. for a supposedly 20 oz. box. Suppose the standard deviation of the amount...
27. ### Defective Tube Estimation Using Central Limit Theorem

Homework Statement The cross sectional area of a tube is u = 12.5 and SD = .2. When the area is less than 12 or greater than 13 it won't work. They are shipped in boxes of 1000, determine how many per box will be defective Homework Equations The Attempt at a Solution So I think...
28. ### Statistics, Parameters, and the Central Limit Theorem

Hi, everyone: I am teaching an intro. stats course, and I want to find a convincing explanation of how we can "reasonably" estimate a population parameter by taking random samples. Given that the course is introductory, I cannot do a proof of the CLT. Specifically, what has seemed...
29. ### Problems applying the central limit theorem

So from what I understand, the central limit theorem allows us to calculate the probability of the mean of a number of independent observations of the same variable. I probably have not understood something because I can't really solve any of the problems just based on the formula give...
30. ### Central Limit Theorem Variation for Chi Square distribution?

Central Limit Theorem Variation for Chi Square distribution? If this question fits into Homework Help, please move it over there. I'm not too sure. I encountered the following problem: Now, this problem seems fairly similar to a simple proof the central limit theorem. I am damn sure...
31. ### Is there a central limit theorem for Median?

Homework Statement Hi, We know the famous central limit theorem for means. I wonder if there is a central limit theorem for Median? If so under what regularity condition, does the median converge to a normal distribution with mean and variance equal to what? Homework Equations The...
32. ### Relationship b/w Binomial, CLT & Poisson Distrib.

From the central limit theorem the binomial distribution can be approximated by a normal distribution N(0,1). But the binomial distribution can also be approximated by a poisson distribition. Does this mean there is a relationship between the normal distribution and the poisson distribution...
33. ### Using TI 89 to apply central limit theorem

(to find distribution of sample mean) Given P((X1 - μ) / σ/√n) < Z < (X2 - μ) / σ/√n)) = P(a < Z < b) = phi(b) - phi(a) where phi(z) = 1/sqrt(2*pi) * integral of exp(-z^2 / 2) dz from negative infinity to z --- I'm sure there's some statistical way of doing this with a TI 89, but...
34. ### Using Central Limit Theorem to Estimate Sample Means in a Stats Class"

Homework Statement Each of 180 students in an evening stats and methods class is asked to generate 64 random numbers with a "spinner" that selects numbers from 1 to 50, and then compute the mean of the 64 numbers. The mean for the class as a whole is 27 with a standard deviation of 20. How...
35. ### Central Limit Theorem Question

Homework Statement On average one third of seniors at a college will be bring parents to the graduation, one third will bring one parent and the remaining third will not bring any parents. Suppose there are 600 seniors graduating this year. Estimate the probability that more than 650 parents...
36. ### Central Limit Theorem Homework: Find Expected Deviation

Homework Statement consider an experiment with 2 possible outcomes, 1 and 0, with a priori probabilities p and 1-p. we would like to find out the average (expected) deviation after N trials, of the relative frequency of the "1"s, N1/N Use the central limit theorem to find expected...
37. ### Is the Central Limit Theorem Applicable to All Random Variables?

i got 2 different answer when i search it.. "The Central Limit Theorem mean of a sampling distribution taken from a single population" is that true for you guys?
38. ### Central Limit Theorem and probability

Homework Statement What is the probability that the average of 150 random points from the interval (0,1) is within .02 of the midpoint of the interval? Homework Equations The Attempt at a Solution I need to determine P(.48<((X1...X150)/150)<.52). I think I need to compute the...
39. ### Calculating Sample Number with Central Limit Theorem for Log-Normal Distribution

Say I have a a log-normal distrubution of data. I want to use the central limit theorem to calculate how big the sample number should be. I would use the geometric standard deviation and we're dealing with a log normal distribution, correct? Using the CLT I can arrive at the equation: n =...
40. ### How Does Sample Size Influence the Variance of the Sample Mean?

Homework Statement If all possible samples of size 16 are drawn from a normal population with mean equal to 50 and standard deviation equal to 5, what is the probability that a sample mean \bar{X} will fall in the interval from \mu_\bar{X} - 1.9 \sigma_\bar{X} to...
41. ### Solving P(11<M<12) Using CLT and Standard Normal Distribution CDF

Homework Statement Let M=(X1+X2+...+X100)/100 where each Xi's has the same distribution of X. Find P(11<M<12) in terms of the cumulative distribution function for the standard normal distribution. Homework Equations The Attempt at a Solution This looks like a "central limit...

please help me with Gaussian Distribution and central limit theorem in matlab! :cry:I am trying to generate a random variable with a approximate gaussian distribution using the rand function and central limit theorem, got stuck when trying it. Please help me. Also want to know how to produce a...
43. ### Estimating Web Server Capacity using Central Limit Theorem

In anyone minute interval, the number of requests for a popular web page is a poisson random variable with expected value 300 requests. a) a web server has a capacity of C requests per minute. if the number of requests in a one minute interval is greater than C, the server is overloaded. Use...
44. ### Weak Law of Large Numbers versus Central Limit Theorem

Hi I was studying the WLLN and the CLT. A form of WLLN states that if X_{n} is a sequence of random variables, it satisfies WLLN if there exist sequences a_{n} and b_{n} such that b_{n} is positive and increasing to infinity such that \frac{S_{n}-a_{n}}{b_{n}} \rightarrow 0 [convergence in...
45. ### Optimizing Magazine Stock Using Probability and the Central Limit Theorem

Homework Statement A newsagent finds that the probability of selling 50n copies of a certain magazine is 2^{-n-1} for n = 1,2,3,... What is the largest sensible number of copies of the magazine that they should stock. Homework Equations The Attempt at a Solution I really don't...
46. ### Probability Histogram and Central Limit Theorem

Hi, I have trouble understanding the convergence of empirical histogram to probability histogram and the convergence of empirical histogram to normal curve. It was written in my lecture notes that as the number of repetitions goes large, empirical histogram converges to probability...
47. ### Central limit theorem and poker

Can you use the central limit theorem to prove that different people have different amount of luck over time in poker? Was a couple of years ago i studied probabilties and statistics so don't really remember how to use it. The theorem most often called the central limit theorem is the...
48. ### Proof of Central Limit Theorem

I have been reading some books about the proof of the Central Limit Theorem, all of them use the uniqueness of moment generating function. But since I have not yet seen a proof of the uniqueness properties, is there any proof that does not use this result? Thanks.
49. C

### Central Limit Theorem and Standardized Sums

I don't think I *really* understand the Central Limit Theorem. Suppose we have a set of n independent random variables \{X_i\} with the same distribution function, same finite mean, and same finite variance. Suppose we form the sum S_n = \sum_{i=1}^n X_i. Suppose I want to know the...
50. ### How Does the Central Limit Theorem Apply to Insurance Risk Assessment?

Hi all, I just wrote a test from probability and had troubles doing this problem: Homework Statement The assurance company makes an insurance for 1000 people of the same age. The probability of death during the year is 0.01 for each of them. Each insured person pays 1.200 dollars a year. In...