weekly outlook

1. Nifty continued it's downtrend for the second consecutive week


2. No fresh longs should be initiated unless the weekly close happens above the BSAR point(Weekly), which now stands at 4868


3. The Weekly Oversold Level is at 4714, any close below this on the first three days of the next week would see a nice bounce of 100 - 150 points, which could be traded


4. The Weekly VHF is at 0.22 almost 2 sd away from the median, signifying that a reversal is at hand


5. The weekly ADX stands at 19, almost one sd away from it's median value, signifying that Nifty is lacking in strong trending oscillations


6. For some strange reason NSE is not publishing the Advance Decline Data for January, so we look at PCR, Put writing is not as feverish as it should be near market bottoms, so there is a possibility that nifty might test Monthly BSAR or Quarterly Mean which is at 4546


7. Some Important Pivots could be, 4920,4883,4861,4790,4766,4742,4719,4695,4671,4647,4620,4576,4554,4517


8. Gold has touched 1050, now the target is 1320.

So where are we?

The FIIs are selling relentlessly, while DII buying is declining. The Nifty is Oversold in weekly charts, there are indications of a possible reversal in longer term charts, but short term reversal signals are absent. A lot of volatility, as noted earlier could be expected, FII, DII data appears courtesy bseindia.com

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fibonacci analysis

1. Nifty made a high of 5310 on 6th Jan 2010, and subsequently made a low of 4766 on 29th of Jan 2010.


2. The difference comes out to be 544 points


3. => 61.8% of 544=336


4. Add 336 to 4766 = 5102


5. As long as Nifty is below 5102, it could be a correction of the 5310 -> 4766 move


6. The ideal corrective target is 4973 (38.2% of 544 added to 4766)


7. If it closes above 5102 the target could be 5457 (127.2% of 544 added to 4766)


8. If it closes below 4766 the target could be 4618 (127.2% of 544 subtracted from 5310)


9. 4894 is a major support now (23.6%), below which nifty would be bearish

Chebyshev’s Inequality (Oh Those Russians)

1. Introduction
   

   Chebyshev's inequality

   
   

   In probability theory, a theorem that characterizes the dispersion of data away from its mean (average). The general theorem is attributed to the 19th-century Russian mathematician Pafnuty Chebyshev, though credit for it should be shared with the French mathematician Irénée-Jules Bienaymé, whose (less general) 1853 proof predated Chebyshev’s by 14 years.
   An observation is rarely more than a few standard deviations away from the mean. Chebyshev's inequality entails the following bounds for all distributions for which the standard deviation is defined.
   
   At least 50% of the values are within √2 standard deviations from the mean.
   At least 75% of the values are within 2 standard deviations from the mean.
   At least 89% of the values are within 3 standard deviations from the mean.
   At least 94% of the values are within 4 standard deviations from the mean.
   At least 96% of the values are within 5 standard deviations from the mean.
   At least 97% of the values are within 6 standard deviations from the mean.
   
   And in general:
   
   At least (1 − 1/k²) × 100% of the values are within k standard deviations from the mean




2. Proof

   Chebyshev's Inequalities
   Chebyshev's inequality and its descendants allow you to place an upper bound on the probability that some random variable is >= a set value, given only the mean and variance of that variable.
   No other information about that variable's distribution is required.
   Some of the descendants exist to make use of information about higher moments, though.

   
   

   This web page was sparked off by a web page from Henry Bottomley, at http://www.btinternet.com/~se16/hgb/cheb.htm. He gives proofs for the standard two-tailed inequality and for a one-tailed variant, and shows examples where the inequality is exactly satisified. Herman Rubin responded in usenet, linking this to the 'problem of moments'. This result is also Question 9 in Problems 7.11 of 'Probability and Random Processes', by Grimmett and Stirzaker, published by Oxford Science Publications, ISBN 0 19 853665 8 - page 327 of my copy, which is the 1993 reprinting. The main contribution of this page is to give a slightly different proof for the one-tailed case, which I find easier to follow than Henry's.

   
   

   The main use for Chebyshev's inequality is in proving theorems, such as laws of large numbers. The existence of cases where it is exact shows that it is as strong as it can be, given its total lack of assumptions on the distribution of the variable involved. Despite this, you can usually get very much stronger bounds by assuming, for instance, that the variable in question is normally distributed - so most people do.
   It would be nice to use Chebyshev's inequalities as a defence against variables in real life where outliers are expected, but in such cases the variance of the variable in question may not be known - and judging the error of estimates of variance will probably involve you in distributional assumptions anyway.

   
   

   The standard two-tailed inequality

   
   

   We are given a variable X with known mean and variance. For convenience, add a constant to ensure that the mean of X is zero. P(|X| >= t) = E([X² >= t²]), where [condition] has the value 1 if condition holds and 0 otherwise. E([X^2 >= t^2]) <= E(X²/t²) = Var(X)/t² (since we made sure E(X) = 0). So P(|X| >= t) <= Var(X)/t². If you remove the constant added you find that P(|X - E(X)| >= t) <= Var(X)/t². This equality can be exact for rather degenerate distributions where X can takes two possible values.

   
   

   A proof of a one-tailed inequality

   
   

   This proof will use the well-known equality Var(X) = E(Var(X|A)) + Var(E(X|A)). One proof of this is:

   Var(X)	=	E(X2) - E(X)2
   	=	E(E(X2|A)) - E(E(X|A))2
   	=	E(Var(X|A) + E(X|A)2) - E(E(X|A))2
   	=	E(Var(X|A)) + E(E(X|A)2) - E(E(X|A))2
   	=	E(Var(X|A)) + Var(E(X|A))

   We first of all subtract a constant as necessary to ensure that E(X) = 0. Now, let p, q, and r be the probabilities that X is > t, = t, and < 0 =" pE(X|X"> t) + qt + rE(X|X < s =" 1"> t, 0 if X = t, and -1 if X <>2) = E(Var(X|S)) + Var(E(X|S))
   And we ignore the term E(Var(X|S)) - which will be zero when our inequality is exact. So we are concerned only with Var(E(X|S)). It's easy to see that E(X|S >= 0) >= t. We can get an idea of E(X|S = -1) from the equality E(X) = 0. Note also that E(E(X|S)) = 0.

   Var(X)	>=	pt2 + qt2 + r[(-pt - qt)/r]2
   Var(X)	>=	(1-r)t2 + t2((1-r)/r)2r
   	>=	t2(1-r) + t2(1-r)2/r
   	>=	t2(1-r)(r+1-r)/r
   	>=	t2s/(1-s)

3. Where s is the prob of X >= t. So Var(X) - sVar(X) >= t²s. or Var(X) >= s(t² + Var(X)), from which we work out that (adding in our constant again)..

   Prob(X - E(X) >= t) <= Var(X)/(t² + Var(X))

   Proof in answer book

   The proof in the answer book is neater, and doesn't need as much inspiration as you might think. Here we assume E(X)=0. Now give yourself an extra degree of freedom by noticing that P(X >= t) = P(X + c > = t + c). Since t + c >= 0, P(X >= t) <= E((X + c)²/(t + c)²) = (Var(X) + c²)/(t + c)².

   If you minimise the rhs over c, you will find that it occurs at c = Var(x)/t. I need only pluck this figure from the air and work out that P(X >= t) <= (Var(X) + Var(X)²/t²)/(t+Var(x)/t)² = Var(X)(t + Var(X)/t)/(t(t+Var(X)/t)²) = Var(X)/(t² + Var(x)).

   Proof in Feller

   Another proof of this appears in Volume II of "An introduction to Probability Theory and its Applications", by Feller, section V.7, example (a). This is similar to the proof in the answer book (but no doubt precedes it, at least if you judge by the date of the respective copyrights.

   Connection with the Median

   The first proof above was based on a proof in Bartoszynski and Niewiadomska-Bugaj that the median is no more than one standard deviation away from the mean, so it is fitting that Henry Bottomley pointed out that you can derive that fact from this inequality. Without loss of generality suppose that the median is greater than the mean. The probablity of X being at least the median value is at least one half. If the median was more than one standard deviation away from the mean, that would contradict the one-tailed Chebyshev inequality.(notes from A.G.McDowell)

4. *****This Topic is Incomplete, would be reposted*****
   

Outlook for February 2010

1. The Overbought level is 5478, the Oversold Level is 4297


2. The Phantom Price Line is at 4888, BSAR is at 4628


3. The Important Pivots would be, 5353,5306,5228,5182,5032,4982,4932,4882,4832,4782,4732,4582,4536,4458,4411


4. The Nifty is buy on dips, keep SL @ 4628


5. The PCR for Nifty Options for January: µ=1.12, s.d.=0.13 σ=1 band =1.25 to 0.99;
   closed on 0.92; we could expect some rampant Put writing now. Also Supports the short term Bullish view


6. In last two three months we have seen historically quiet periods, but don't be mistaken


7. All weekly indicators are near their lower sigma

the central limit theorem

In probability theory, the central limit theorem (CLT) states conditions under which the mean of a sufficiently large number of independent random variables, each with finite mean and variance, will be approximately normally distributed (Rice 1995). The central limit theorem also requires the random variables to be identically distributed, unless certain conditions are met. Since real-world quantities are often the balanced sum of many unobserved random events, this theorem provides a partial explanation for the prevalence of the normal probability distribution. The CLT also justifies the approximation of large-sample statistics to the normal distribution in controlled experiments.

In more general probability theory, a central limit theorem is any of a set of weak-convergence theories. They all express the fact that a sum of many independent random variables will tend to be distributed according to one of a small set of "attractor" (i.e. stable) distributions. For other generalizations for finite variance which do not require identical distribution, see Lindeberg's condition, Lyapunov's condition, Gnedenko and Kolmogorov states.

The central limit theorem states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases. The amazing and counter-intuitive thing about the central limit theorem is that no matter what the shape of the original distribution, the sampling distribution of the mean approaches a normal distribution. Furthermore, for most distributions, a normal distribution is approached very quickly as N increases. Keep in mind that N is the sample size for each mean and not the number of samples. Remember in a sampling distribution the number of samples is assumed to be infinite. The sample size is the number of scores in each sample; it is the number of scores that goes into the computation of each mean.

Courtesy: Wikipedia.com, davidmlane.com