Misplaced Pages

#339660

64-469: Premium Bonds is a lottery bond scheme organised by the United Kingdom government since 1956. At present it is managed by the government's National Savings and Investments agency. The principle behind Premium Bonds is that rather than the stake being gambled, as in a usual lottery , it is the interest on the bonds that is distributed by a lottery. The bonds are entered in a monthly prize draw and

128-447: A ↦ E ⁡ ( ‖ X − a ‖ ) . {\displaystyle a\mapsto \operatorname {E} (\|X-a\|).\,} The spatial median is unique when the data-set's dimension is two or more. An alternative proof uses the one-sided Chebyshev inequality; it appears in an inequality on location and scale parameters . This formula also follows directly from Cantelli's inequality . For

192-624: A bond that earns no interest but is eligible for entry into a lottery. The modern iteration of Premium Bonds were introduced by Harold Macmillan , as Chancellor of the Exchequer , in his Budget of 17 April 1956, to control inflation and encourage people to save. On 1 November 1956, in front of the Royal Exchange in the City of London, the Lord Mayor of London , Alderman Sir Cuthbert Ackroyd , bought

256-763: A probability density function f ), nor does it require a discrete one . In the former case, the inequalities can be upgraded to equality: a median satisfies P ⁡ ( X ≤ m ) = ∫ − ∞ m f ( x ) d x = 1 2 {\displaystyle \operatorname {P} (X\leq m)=\int _{-\infty }^{m}{f(x)\,dx}={\frac {1}{2}}} and P ⁡ ( X ≥ m ) = ∫ m ∞ f ( x ) d x = 1 2 . {\displaystyle \operatorname {P} (X\geq m)=\int _{m}^{\infty }{f(x)\,dx}={\frac {1}{2}}\,.} Any probability distribution on

320-404: A comment on a subsequent proof by O'Cinneide, Mallows in 1991 presented a compact proof that uses Jensen's inequality twice, as follows. Using |·| for the absolute value , we have The first and third inequalities come from Jensen's inequality applied to the absolute-value function and the square function, which are each convex. The second inequality comes from the fact that a median minimizes

384-523: A default due to the United States debt ceiling . In 2008, two financial economists, Lobe and Hoelzl, analysed the main driving factors for the immense marketing success of Premium Bonds. One in three Britons invest in Premium Bonds. The thrill of gambling is significantly boosted by enhancing the skewness of the prize distribution. However, using data collected over the past fifty years, they found that

448-401: A fixed, though usually long, duration and either pay no interest or regular coupons . The individual bonds within each issue are numbered, like ordinary bonds, but the serial numbers serve a different function from ordinary bonds. For a lottery bond the serial number is an added incentive for the purchaser to buy the bond. Although the details vary by bond and by issuer , the principle remains

512-399: A half hours to complete its monthly draw. In August 2004, ERNIE 4 was brought into service in anticipation of an increase in prizes each month from September 2004. Developed by LogicaCMG , it was 500 times faster than the original and generated a million numbers an hour; these were checked against a list of valid bonds. By comparison, the original ERNIE generated 2,000 numbers an hour and was

576-408: A median is based on the middle data in a set, it is not necessary to know the value of extreme results in order to calculate it. For example, in a psychology test investigating the time needed to solve a problem, if a small number of people failed to solve the problem at all in the given time a median can still be calculated. Because the median is simple to understand and easy to calculate, while also

640-430: A median is defined as any real number  m that satisfies the inequalities lim x → m − F ( x ) ≤ 1 2 ≤ F ( m ) {\displaystyle \lim _{x\to m-}F(x)\leq {\frac {1}{2}}\leq F(m)} (cf. the drawing in the definition of expected value for arbitrary real-valued random variables ). An equivalent phrasing uses

704-428: A median of a population is any value such that at least half of the population is less than or equal to the proposed median and at least half is greater than or equal to the proposed median. As seen above, medians may not be unique. If each set contains more than half the population, then some of the population is exactly equal to the unique median. The median is well-defined for any ordered (one-dimensional) data and

SECTION 10

#1732845599340

768-452: A median value of 4.5 , that is ( 4 + 5 ) / 2 {\displaystyle (4+5)/2} . (In more technical terms, this interprets the median as the fully trimmed mid-range ). In general, with this convention, the median can be defined as follows: For a data set x {\displaystyle x} of n {\displaystyle n} elements, ordered from smallest to greatest, Formally,

832-501: A random variable X distributed according to F : P ⁡ ( X ≤ m ) ≥ 1 2  and  P ⁡ ( X ≥ m ) ≥ 1 2 . {\displaystyle \operatorname {P} (X\leq m)\geq {\frac {1}{2}}{\text{ and }}\operatorname {P} (X\geq m)\geq {\frac {1}{2}}\,.} Note that this definition does not require X to have an absolutely continuous distribution (which has

896-410: A robust approximation to the mean , the median is a popular summary statistic in descriptive statistics . In this context, there are several choices for a measure of variability : the range , the interquartile range , the mean absolute deviation , and the median absolute deviation . For practical purposes, different measures of location and dispersion are often compared on the basis of how well

960-424: A single pass over the sample. The distributions of both the sample mean and the sample median were determined by Laplace . The distribution of the sample median from a population with a density function f ( x ) {\displaystyle f(x)} is asymptotically normal with mean μ {\displaystyle \mu } and variance where m {\displaystyle m}

1024-473: A well-defined mean, such as the Cauchy distribution : The mean absolute error of a real variable c with respect to the random variable   X is Provided that the probability distribution of X is such that the above expectation exists, then m is a median of X if and only if m is a minimizer of the mean absolute error with respect to X . In particular, if m is a sample median, then it minimizes

1088-566: A windfall. The earliest recorded lottery bond in UK was in 1694, called Million Lottery , where people purchased lottery tickets (priced £1 million in total) that doubled as a bond that matured in 16 years. Equivalently, they were bonds with annual interest rate 10+x%, where x is drawn from a lottery that gave x = 1.5 effectively. The tickets themselves remained a tradable commodity after the lottery draw (in November 1694), because each ticket still yielded £1

1152-428: A year compared to 15 the previous year. Investors with smaller, although significant, amounts would possibly win nothing. From 1 January 2009 the odds of winning a prize for each £1 of bond was 36,000 to 1. In October 2009, the odds returned to 24,000 to 1 with the prize fund interest rate increase. The odds reached 26,000 to 1 by October 2013 and then reverted to 24,500 to 1 in November 2017. As of September 2023,

1216-517: A year for 16 years. The price of a ticket fluctuated around £5-8 during 1694-1701. This proved popular and the government repeated this until 1769. The government of the UK offers a variation on the standard Lottery Bond. Through the NS&;I (National Savings and Investment), the public can purchase Premium Bonds worth £1 each, with a minimum spend of £25. The maximum number of Bonds that an individual can hold

1280-411: A £50,000 cap – or can be paid out. All prizes are tax free and, with approximately 84 billion bonds issued, the chances of any one bond winning a prize for a given month are approximately 24500 to 1. However, if a bond wins a prize, that bond is not redeemed but remains 'in the pool' for all forthcoming draws (at least until the bond-holder decides to redeem it.). The prize fund is paid for out of

1344-404: Is £ 50,000. The bonds themselves attract no interest, are perpetual and are redeemable at par (face value) at any time. The attraction for an investor is that, each month, a draw takes place and, should an investor hold one of the bond numbers chosen, then the bond-holder will be awarded a prize of variable value. Prizes (which vary between £25 and £1 million) can be reinvested – subject to

SECTION 20

#1732845599340

1408-469: Is a C function, but the reverse does not hold. If f is a C function, then If the medians are not unique, the statement holds for the corresponding suprema. Even though comparison-sorting n items requires Ω ( n log n ) operations, selection algorithms can compute the k th-smallest of n items with only Θ( n ) operations. This includes the median, which is the ⁠ n / 2 ⁠ th order statistic (or for an even number of samples,

1472-465: Is a special case of other ways of summarizing the typical values associated with a statistical distribution : it is the 2nd quartile , 5th decile , and 50th percentile . The median can be used as a measure of location when one attaches reduced importance to extreme values, typically because a distribution is skewed , extreme values are not known, or outliers are untrustworthy, i.e., may be measurement or transcription errors. For example, consider

1536-425: Is equal to 150,000 EUR and a further 9,880 at 1,000 EUR (9,880,000 EUR) making a total of 10,030,000 Euros, or 100.3% of the original borrowing. The issuer will also pay interest on any unredeemed bonds. Lottery bonds are usually issued in a period where investor zeal is low and the government may see an issue failing to sell. By knowing ahead of time when the coupons will be paid and how many bonds will be redeemed at

1600-400: Is independent of any distance metric . The median can thus be applied to school classes which are ranked but not numerical (e.g. working out a median grade when student test scores are graded from F to A), although the result might be halfway between classes if there is an even number of classes. (For odd number classes, one specific class is determined as the median.) A geometric median , on

1664-424: Is less than the mean, as shown in the figure. Jensen's inequality states that for any random variable X with a finite expectation E [ X ] and for any convex function f This inequality generalizes to the median as well. We say a function f : R → R is a C function if, for any t , is a closed interval (allowing the degenerate cases of a single point or an empty set ). Every convex function

1728-640: Is now at 4.65% from September 2023. The following table lists the distribution of prizes on offer in the September 2023 draw. While the mean return is 4.4% as of March 2024, the median return is lower. For an investor with the maximum £50,000 invested, the median return is 3.9% (£1,950). For investors with lower amounts invested, the median return is lower. The typical investor with £1000 or less invested will receive nothing. Premium Bonds are tax free, so are more attractive to higher rate taxpayers. ERNIE - an acronym for "Electronic Random Number Indicator Equipment" -

1792-520: Is the median of f ( x ) {\displaystyle f(x)} and n {\displaystyle n} is the sample size: A modern proof follows below. Laplace's result is now understood as a special case of the asymptotic distribution of arbitrary quantiles . For normal samples, the density is f ( m ) = 1 / 2 π σ 2 {\displaystyle f(m)=1/{\sqrt {2\pi \sigma ^{2}}}} , thus for large samples

1856-539: Is the name for a series of hardware random number generators developed for this application. There have been five models of ERNIE to date. All of them have generated true random numbers derived from random statistical fluctuations in a variety of physical processes. The first ERNIE was built at the Post Office Research Station by a team led by Sidney Broadhurst. The designers were Tommy Flowers and Harry Fensom and it derives from Colossus , one of

1920-401: Is uncontaminated by data from heavy-tailed distributions or from mixtures of distributions. Even then, the median has a 64% efficiency compared to the minimum-variance mean (for large normal samples), which is to say the variance of the median will be ~50% greater than the variance of the mean. For any real -valued probability distribution with cumulative distribution function   F ,

1984-410: The absolute deviation function a ↦ E ⁡ ( | X − a | ) {\displaystyle a\mapsto \operatorname {E} (|X-a|)} . Mallows's proof can be generalized to obtain a multivariate version of the inequality simply by replacing the absolute value with a norm : where m is a spatial median , that is, a minimizer of the function

Premium Bonds - Misplaced Pages Continue

2048-422: The arithmetic mean of the two middle order statistics). Selection algorithms still have the downside of requiring Ω( n ) memory, that is, they need to have the full sample (or a linear-sized portion of it) in memory. Because this, as well as the linear time requirement, can be prohibitive, several estimation procedures for the median have been developed. A simple one is the median of three rule, which estimates

2112-498: The multiset The median is 2 in this case, as is the mode , and it might be seen as a better indication of the center than the arithmetic mean of 4, which is larger than all but one of the values. However, the widely cited empirical relationship that the mean is shifted "further into the tail" of a distribution than the median is not generally true. At most, one can say that the two statistics cannot be "too far" apart; see § Inequality relating means and medians below. As

2176-525: The UK was the football pools , with the National Lottery not coming into existence until 1994. Although many avenues of lotteries and other forms of gambling are now available to British adults, Premium Bonds are held by more than 24 million people, equivalent to more than 1 in 3 of the UK population. The term "premium bond" has been used in the English language since at least the late 18th century, to mean

2240-453: The arithmetic mean of the absolute deviations. Note, however, that in cases where the sample contains an even number of elements, this minimizer is not unique. More generally, a median is defined as a minimum of as discussed below in the section on multivariate medians (specifically, the spatial median ). This optimization-based definition of the median is useful in statistical data-analysis, for example, in k -medians clustering . If

2304-531: The bond bears relatively low risk compared to many other investments. Aaron Brown discusses in a 2006 book Premium Bonds in comparison with equity-linked , commodity-linked and other "added risk" bonds. His conclusion is that it makes little difference, either to a retail investor or from a theoretical finance perspective, whether the added risk comes from a random number generator or from fluctuations in financial markets. Lottery bond Outwardly, lottery bonds resemble ordinary fixed rate bonds ; they have

2368-449: The bond's maturity date and has further committed to redeem 120 of these at a redemption value of EUR 1,250. This means that any purchaser of a single bond for EUR 1,000 will receive annual interest a little above the bank rate, but will also have a 1.2% chance of winning an additional 25% of their original investment. So the issuer borrows 10,000 * 1,000 EUR which is 10 Million Euros . It will repay 120 of those bonds at 1,250 EUR which

2432-576: The case of unimodal distributions, one can achieve a sharper bound on the distance between the median and the mean: A similar relation holds between the median and the mode: A typical heuristic is that positively skewed distributions have mean > median. This is true for all members of the Pearson distribution family . However this is not always true. For example, the Weibull distribution family has members with positive mean, but mean < median. Violations of

2496-401: The center. Median income , for example, may be a better way to describe the center of the income distribution because increases in the largest incomes alone have no effect on the median. For this reason, the median is of central importance in robust statistics . The median of a finite list of numbers is the "middle" number, when those numbers are listed in order from smallest to greatest. If

2560-436: The corresponding population values can be estimated from a sample of data. The median, estimated using the sample median, has good properties in this regard. While it is not usually optimal if a given population distribution is assumed, its properties are always reasonably good. For example, a comparison of the efficiency of candidate estimators shows that the sample mean is more statistically efficient when—and only when— data

2624-438: The data set has an odd number of observations, the middle one is selected (after arranging in ascending order). For example, the following list of seven numbers, has the median of 6 , which is the fourth value. If the data set has an even number of observations, there is no distinct middle value and the median is usually defined to be the arithmetic mean of the two middle values. For example, this data set of 8 numbers has

Premium Bonds - Misplaced Pages Continue

2688-412: The distribution has finite variance, then the distance between the median X ~ {\displaystyle {\tilde {X}}} and the mean X ¯ {\displaystyle {\bar {X}}} is bounded by one standard deviation . This bound was proved by Book and Sher in 1979 for discrete samples, and more generally by Page and Murty in 1982. In

2752-416: The draw each month, with an equal chance of winning, until the bond is cashed. As of 2019, each person may own bonds up to £50,000. Since 1 February 2019, the minimum purchase amount for Premium Bonds has been £25. As of September 2023 there are over 121   billion eligible Premium Bonds, each having a value of £1 . When introduced to the wider public in 1957, the only other similar game available in

2816-406: The equivalent interest payable on the entire bond pool for that month. As of 2020 the prize fund rate is 4.65% implying that a bond holder can expect to achieve a mean long term return of 4.65% per annum. In reality, the nature of a lottery bond means that median returns are lower and are increasing in the number of bonds held. Median The median of a set of numbers is the value separating

2880-524: The first bond from the Postmaster General, Dr Charles Hill , for £1. Councillor William Crook, the mayor of Lytham St Anne's , bought the second. The Premium Bonds office was in St Annes-on-Sea , Lancashire, until it moved to Blackpool in 1978. Winners of the jackpot are told on the first working day of the month, although the actual date of the draw varies. The online prize finder is updated by

2944-470: The formula for the case of discrete variables is given below in § Empirical local density . The sample can be summarized as "below median", "at median", and "above median", which corresponds to a trinomial distribution with probabilities F ( v ) {\displaystyle F(v)} , f ( v ) {\displaystyle f(v)} and 1 − F ( v ) {\displaystyle 1-F(v)} . For

3008-455: The government promises to buy them back, on request, for their original price. The government pays interest into the bond fund (4.4% per annum from March 2024) from which a monthly lottery distributes tax-free prizes to bondholders whose numbers are selected randomly. The machine that generates the numbers is called ERNIE , an acronym for "Electronic Random Number Indicator Equipment". Prizes range from £25 to £1,000,000 and (since September 2023)

3072-417: The higher half from the lower half of a data sample , a population , or a probability distribution . For a data set , it may be thought of as the “middle" value. The basic feature of the median in describing data compared to the mean (often simply described as the "average") is that it is not skewed by a small proportion of extremely large or small values, and therefore provides a better representation of

3136-520: The latest model, was brought into service in March 2019, and is a quantum random number generator built by ID Quantique . It uses quantum technology to produce random numbers through light, replacing the former 'thermal noise' method. Running at speeds 21,000 times faster than the first ERNIE, it can produce 3 million winners in just 12 minutes each month. ERNIE, anthropomorphised in early advertising, receives Valentine cards, Christmas cards and letters from

3200-459: The median as the median of a three-element subsample; this is commonly used as a subroutine in the quicksort sorting algorithm, which uses an estimate of its input's median. A more robust estimator is Tukey 's ninther , which is the median of three rule applied with limited recursion: if A is the sample laid out as an array , and then The remedian is an estimator for the median that requires linear time but sub-linear memory, operating in

3264-495: The odds of a £1 bond winning a prize in a given month are 21,000 to 1. Investors can buy bonds at any time but they must be held for a whole calendar month before they qualify for a prize. As an example, a bond purchased mid-May must then be held throughout June before being eligible for the draw in July (and onwards). Bonds purchased by reinvestment of prizes are immediately eligible for the following month's draw. Numbers are entered in

SECTION 50

#1732845599340

3328-450: The odds of winning are 1/21000; resulting in the expected number of prizes for the maximum £50,000 worth of bonds being 29 per year. The prize fund is equal to one month's interest on all bonds eligible for the draw. The annual interest is set by NS&I and was 1.40% as of December 2017, reducing to 1.00% as of December 2020. This was increased to 2.2%, as of October 2022 then increased again to 3% as of January 2023 and

3392-401: The original value and at the lottery value, the issuer can value the bond accurately and know ahead of time the cost of the borrowing. The purchaser, however, is subject to the element of chance, as it is not known in advance which actual bonds will be redeemed at either value, nor when. This element of chance appeals to a section of society who will take a lower guaranteed return in the hope of

3456-474: The other hand, is defined in any number of dimensions. A related concept, in which the outcome is forced to correspond to a member of the sample, is the medoid . There is no widely accepted standard notation for the median, but some authors represent the median of a variable x as med( x ), x͂ , as μ 1/2 , or as M . In any of these cases, the use of these or other symbols for the median needs to be explicitly defined when they are introduced. The median

3520-439: The public. It is the subject of the song "E.R.N.I.E." by Madness , from the 1980 album Absolutely . It is also referenced by Jethro Tull in their album Thick as a Brick . Premium Bonds under various names exist or have existed in various countries. Similar programmes to UK Premium Bonds include: In 2023, American economist Paul Krugman used the name "premium bonds" for an unrelated type of bond that he proposed to avoid

3584-430: The real number set R {\displaystyle \mathbb {R} } has at least one median, but in pathological cases there may be more than one median: if F is constant 1/2 on an interval (so that f = 0 there), then any value of that interval is a median. The medians of certain types of distributions can be easily calculated from their parameters; furthermore, they exist even for some distributions lacking

3648-400: The rule are particularly common for discrete distributions. For example, any Poisson distribution has positive skew, but its mean < median whenever μ mod 1 > ln ⁡ 2 {\displaystyle \mu {\bmod {1}}>\ln 2} . See for a proof sketch. When the distribution has a monotonically decreasing probability density, then the median

3712-442: The same. A drawing takes place according to a schedule to decide which serial numbers are to be redeemed. The individual bonds within the issue thus identified by the drawing are then bought back by the issuer, so that the total value of an issue will decrease as time passes and more bonds are redeemed. A small number of bonds are redeemed for an amount greater than their face value. Hence the holder of that particular bond will have won

3776-471: The size of a van. ERNIE 4 used thermal noise in transistors as its source of randomness to generate true random numbers. ERNIE's output was independently tested each month by the Government Actuary's Department , the draw being valid only if it was certified to be statistically consistent with randomness. At the end of its life it was moved to Bletchley Park's National Museum of Computing . ERNIE 5,

3840-814: The smartphone app, which provides lists of winning bond numbers for the past six months. Older winning numbers (more than 18 months old) can also be checked in the London Gazette Premium Bonds Unclaimed Prizes Supplement . In December 2008, NS&I reduced the interest rate (and therefore the odds of winning) due to the drop in the Bank of England base rate during the Great Recession , leading to criticism from members of Parliament, financial experts and holders of bonds; many claimed Premium Bonds were now "worthless", and somebody with £30,000 invested and "average luck" would win only 10 prizes

3904-680: The third or fourth working day of the month. Winners of the top £1m prize are told in person of their win by "Agent Million", an NS&I employee, usually on the day before the first working day of the month. However, in-person visits were suspended, starting in May 2020, during the COVID-19 pandemic in the United Kingdom . Bond holders can check whether they have won any prizes on the National Savings & Investment Premium Bond Prize Checker website, or

SECTION 60

#1732845599340

3968-420: The variance of the median equals ( π / 2 ) ⋅ ( σ 2 / n ) . {\displaystyle ({\pi }/{2})\cdot (\sigma ^{2}/n).} (See also section #Efficiency below.) We take the sample size to be an odd number N = 2 n + 1 {\displaystyle N=2n+1} and assume our variable continuous;

4032-534: The world's first digital computers. It was introduced in 1957, with the first draw on 1 June, and generated bond numbers from the signal noise created by neon gas discharge tubes . ERNIE 1 is in the collections of the Science Museum in London and was on display between 2008 and 2015. ERNIE 2 replaced the first ERNIE in 1972. ERNIE 3 in 1988 was the size of a personal computer; at the end of its life it took five and

4096-456: The ‘lottery’. Lottery bonds are similar to prize-linked savings accounts . If the government of Belgium issues a 10-year lottery bond to finance re-gilding the Atomium , the issue may consist of 10,000 individual bonds with a face value of EUR 1,000. The coupon rate is reasonable to attract investors, but not high. However, the issuer has committed to redeem 5,000 of the 10,000 bonds issued before

#339660