We have in VS .NET Creation Code 128C in VS .NET We have

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We have using .net framework tointegrate qr barcode with web,windows application Basice Knowlege of iReport peA). = p, P(not A) = q. We now want to see wha Visual Studio .NET QR-Code t happens when we take many repeated trials. The following theorem is key:.

Theorem 15.5.1 (Centra l Limit Theorem) Consider a sample space w = {A, not A} with peA) = p and P(not A) = 1 - p = q.

Given n independent random variables Xl, ...

, X n , each taking. Then for any real numb VS .NET Quick Response Code ers a and b, lim P{a ::;. n---+oo ::; b}. 1 V 211". e-2-dx. _,,2. 15.5. CENTRAL LIMIT THEOREM What this is saying is that if we perform a huge number of repeated Bernoulli trials, then the values of Sn will be distributed as:. But we have even more. .net framework QR Code Namely, by normalizing Sn to the new random variable S~ (which, as we will see in a moment, has mean zero and variance one), we always get the same distribution, no matter what the real world situation we start with is, just as long as the real world problem can be modelled as a Bernoulli trial.

By the way, the distribution for any Bernoulli trial is simply the graph of the function limn~oo Sn. We call S~ the normal distribution. Its graph is the Gauss-Bell curve.

. Before sketching a pro of of the Central Limit Theorem (whose general outline is from [18]), let us look at the random variables Sn and S~.. Lemma 15.5.1 The expec qrcode for .

NET ted value of Sn is np and its variance is npq. The expected value of S~ is a and its variance is 1. Proof of Lemma: We know that for all k,.

E(X k ). = Xk(A)P(A) + Xk(not A)P(not A) = 1 p + O q = p. CHAPTER 15. COMBINATORICS AND PROBABILITY Then by the linearity of the expected value function E(X 1 + E(X 1 ) + +X n ) + E(X n ). As for the variance, w e know that for any k,. E(X~) - [E(X k)]2. X~(A)P(A). p_ p2 + X~(not A)P(not A) _ p2 12 . P + 02 . q _ p2 p(l - p). Then we have V(X 1 + V(X 1 ) + npq. +Xn ) + V(X n ). E(S~). which, since E(Sn) is just a number, is zero. Now for the variance. First, note that for any random variable that happens to be a constant function, the variance must be zero.

In particular, since the expected value of a random variable is a number, we must have that the variance of an expected value is zero:. V(E(X)) = O. Using this, we have that V(S~). 15.5. CENTRAL LIMIT THEOREM (~)2V(Sn1,. E(Sn)) V(Sn) 1 V(Sn) ( V(Sn) - V(E(Sn))). as desired. 0 Before d VS .NET QR iscussing the proof of the Central Limit Theorem, let us look at the formula lim P(a:5:.

S~ :5:. b) = . f(C.

n~oo y27l". jb e--dx. It happens to be the c .net framework qr bidimensional barcode ase that for any particular choice of a and b, it is im-. e _;2 dx; instead peop le possible to explicitly calculate the integral ~ must numerically approximate the answers, which of course can easily be done with standard software packages like Maple or Mathematica. Surprisingly enough, ~ J~oo e -; dx can be shown to be exactly one. We first show why this must be the case if the Central Limit Theorem is true and then we will explicitly prove that this integral is one.

For any sequence of events and for any n, S~ must be some number. Thus for all n, P(-oo :5:. S~ ::; (0) = 1, and thus its limit as n goes to infinity must be one, meaning that our integral is one.

Thus if ~ J~oo eT dx is not one, the Central Limit Theorem would not be true. Thus we need to prove that this integral is one. In fact, the proof that this integral is one is interesting in its own right.

Theorem 15.5.2.

Joo e--dx = 1. _~2 y27l". Proof: Surprisingly, w visual .net Quick Response Code e look at the square of the integral: 1 (--. ..,fFff - 00.

Joo e--dx)2 = ( -1- Joo e--dx)(-- Joo e--dx) . _~2 _~2 1 _~2 2 2 2. ..,fFff ..,fFff - 00.

Since the symbol x jus QR Code for .NET t denotes what variable we are integrating over, we can change the x in the second integral to a y without changing the equality: 1 (-. ..,fFff - 00.

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