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In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of random variables deviates from its expected value. Hoeffding's inequality was proved by Wassily Hoeffding in 1963.

Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality, and it is more general than the Bernstein inequality, proved by Sergei Bernstein in 1923. They are also special cases of McDiarmid's inequality.

Special case of Bernoulli random variables

Hoeffding's inequality can be applied to the important special case of identically distributed Bernoulli random variables, and this is how the inequality is often used in combinatorics and computer science. We consider a coin that shows heads with probability $p$ and tails with probability $1-p$. We toss the coin $n$ times. The expected number of times the coin comes up heads is $p\cdot n$. Furthermore, the probability that the coin comes up heads at most $k$ times can be exactly quantified by the following expression:

$\Pr {\Big (}n{\text{ coin tosses yield heads at most }}k{\text{ times}}{\Big )}=\sum _{i=0}^{k}{\binom {n}{i}}p^{i}(1-p)^{n-i}\,.$

In the case that $k=(p-\epsilon )n$ for some $\epsilon >0$, Hoeffding's inequality bounds this probability by a term that is exponentially small in $\epsilon ^{2}\cdot n$:

$\Pr {\Big (}n{\text{ coin tosses yield heads at most }}(p-\epsilon )n{\text{ times}}{\Big )}\leq \exp {\big (}-2\epsilon ^{2}n{\big )}\,.$

Similarly, in the case that $k=(p+\epsilon )n$ for some $\epsilon >0$, Hoeffding's inequality bounds the probability that we see at least $\epsilon n$ more tosses that show heads than we would expect:

$\Pr {\Big (}n{\text{ coin tosses yield heads at least }}(p+\epsilon )n{\text{ times}}{\Big )}\leq \exp {\big (}-2\epsilon ^{2}n{\big )}\,.$

Hence Hoeffding's inequality implies that the number of heads that we see is concentrated around its mean, with exponentially small tail.

$\Pr {\Big (}n{\text{ coin tosses yield heads between }}(p-\epsilon )n{\text{ and }}(p+\epsilon )n{\text{ times}}{\Big )}\geq 1-2\exp {\big (}-2\epsilon ^{2}n{\big )}\,.$

General case

Let

$X_{1},\dots ,X_{n}\!$

be independent random variables. Assume that the $X_{i}$ are almost surely bounded; that is, assume for $1\leq i\leq n$ that

$\Pr(X_{i}\in [a_{i},b_{i}])=1.\!$

We define the empirical mean of these variables

${\overline {X}}={\frac {1}{n}}(X_{1}+\cdots +X_{n}).$

Theorem 2 of Template:Harvtxt proves the inequalities

$\Pr({\overline {X}}-\mathrm {E} [{\overline {X}}]\geq t)\leq \exp \left(-{\frac {2n^{2}t^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!$

$\Pr(|{\overline {X}}-\mathrm {E} [{\overline {X}}]|\geq t)\leq 2\exp \left(-{\frac {2n^{2}t^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!$

which are valid for positive values of t. Here $\mathrm {E} [{\overline {X}}]$ is the expected value of ${\overline {X}}$. The inequalities can be also stated in terms of the sum

$S=X_{1}+\cdots +X_{n}$

of the random variables:

$\Pr(S-\mathrm {E} [S]\geq t)\leq \exp \left(-{\frac {2t^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right),\!$

$\Pr(|S-\mathrm {E} [S]|\geq t)\leq 2\exp \left(-{\frac {2t^{2}}{\sum _{i=1}^{n}(b_{i}-a_{i})^{2}}}\right).\!$

Note that the inequalities also hold when the $X_{i}$ have been obtained using sampling without replacement; in this case the random variables are not independent anymore. A proof of this statement can be found in Hoeffding's paper. For slightly better bounds in the case of sampling without replacement, see for instance the paper by Template:Harvtxt.

See also

• Bennett's inequality
• Chebyshev's inequality
• Markov's inequality
• Chernoff bounds
• Hoeffding's lemma

Notes

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References

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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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