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In [[probability and statistics]], a '''mean-preserving spread (MPS)'''<ref>[[Michael Rothschild|Rothschild, Michael]], and [[Joseph Stiglitz|Stiglitz, Joseph]], "Increasing risk I: A definition," ''Journal of Economic Theory'', 1970, 225&ndash;243.</ref> is a change from one [[probability distribution]] A to another probability distribution B, where B is formed by spreading out one or more portions of A's [[probability density function]] or [[probability mass function]] while leaving the mean (the [[expected value]]) unchanged.  As such, the concept of mean-preserving spreads provides a [[stochastic ordering]] of equal-mean gambles (probability distributions) according to their degree of [[risk]]; this ordering is [[Partially ordered set|partial]], meaning that of two equal-mean gambles, it is not necessarily true that either is a mean-preserving spread of the other. A is said to be a '''mean-preserving contraction''' of&nbsp;B if B is a mean-preserving spread of A.
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Ranking gambles by mean-preserving spreads is a special case of ranking gambles by second-order [[stochastic dominance]] &ndash; namely, the special case of equal means:  If B is a mean-preserving spread of A, then A is second-order stochastically dominant over B; and the [[contraposition#Examples|converse]] holds if A and B have equal means.
 
If B is a mean-preserving spread of A, then B has a higher variance than A; but the converse is not in general true, because the variance is a complete ordering while ordering by mean-preserving spreads is only partial.
 
==Example==
 
This example from <ref>Landsberger, M., and Meilijson, I., "Mean-preserving portfolio dominance," ''Review of Economic Studies'' 60, April 1993, 479&ndash;485.</ref> shows that to have a mean-preserving spread does not require that all or most of the probability mass move away from the mean.  Let A have equal probabilities <math>1/100</math> on each outcome <math>x_{Ai}</math> , with <math>x_{Ai}=198</math> for <math>i=1,\dots, 50</math> and <math>x_{Ai}=202</math> for <math>i=51,\dots,100</math>; and let B have equal probabilities <math>1/100</math> on each outcome <math>x_{Bi}</math>, with <math>x_{B1}=100</math>, <math>x_{Bi}=200</math> for <math>i=2,\dots,99</math>, and <math>x_{B100}=300</math>.  Here B has been constructed from A by moving one chunk of 1% probability from 198 to 100 and moving 49 probability chunks from 198 to 200, and then moving one probability chunk from 202 to 300 and moving 49 probability chunks from 202 to 200.  This sequence of two mean-preserving spreads is itself a mean-preserving spread, despite the fact that 98% of the probability mass has moved to the mean (200).
 
==Mathematical definitions==
 
Let <math>x_A</math> and <math>x_B</math> be the random variables associated with gambles A and B. Then B is a mean-preserving spread of A if and only if <math>x_B \overset {d}{=} (x_A + z)</math>  for some random variable <math>z</math> having <math> E(z\mid x_A)=0</math> for all values of <math>x_A</math>.  Here <math> \overset{d}{=}</math> means "[[Random_variable#Equality_in_distribution|is equal in distribution to]]" (that is, "has the same distribution as").
 
Mean-preserving spreads can also be defined in terms of the [[cumulative distribution function]]s <math>F_A</math> and <math>F_B</math> of A and B.  If A and B have equal means, B is a mean-preserving spread of A if and only if the area under <math>F_A</math> from minus infinity to <math>x</math> is less than or equal to that under <math>F_B</math> from minus infinity to <math>x</math> for all real numbers <math>x</math>, with strict inequality at some <math>x</math>.
 
Both of these mathematical definitions replicate those of second-order stochastic dominance for the case of equal means.
 
==Relation to expected utility theory==
 
If B is a mean-preserving spread of A then A will be preferred by all [[expected utility hypothesis|expected utility]] maximizers having concave utility. The converse also holds: if A and B have equal means and A is preferred by all expected utility maximizers having concave utility, then B is a mean-preserving spread of A.
 
==See also==
*[[Stochastic ordering]]
 
==References==
 
<references/>
 
{{DEFAULTSORT:Mean-Preserving Spread}}
 
[[Category:Theory of probability distributions]]
[[Category:Decision theory]]

Latest revision as of 19:28, 29 March 2014

Acquiring just what will be the excellent knife to employ when preparing meals depends primarily to the goal and also to the person who tends to make use of it. A single can locate also basic qualities that ought to be taken into account when having these cooking equipments. An additional excellent knife that is a should have amongst the Japanese chef knives is the Santoku knife which is a common all goal knife applied in chopping up larger vegetables such as potatoes, cabbages and pumpkins. Japanese chef's knives are Kitchen Knife Reviews Canada renowned for their higher quality steel building which offers for enhanced uniformity in cutting, sharpness and swiftness - the ingredients for excellence in the kitchen. They also make a imply set of butterfly knives.



In order to keep the knives sharp and not damage them your cutting board requires to be either an Epicuran, bamboo or other hardwood cutting board e.g. maple or cherry. I am a culinary student and decided to go à la carte and mix andknives from each value level. This is not to say that I want to spend a fortune on all my knives. With all the Shun knives you are going to get exceptional steel and higher performance design.

The Wusthof Classic three-1/two-Inch Paring Knife is an additional of those knives that is worth spending a little bit of revenue 'll use it virtually each day and with the lifetime guarantee this knife investment is revenue seriously properly spent. The Wusthof paring knife has a small pointed blade which tends to make it 1 of the best knives for peeling vegetables and for all smaller cutting jobs Kitchen Knives Reviews 2012 Uk in the kitchen.

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On the flip side, far also quite a few individuals make do with poor good quality knives, when this is the singular most significant kitchen tool. If you do not but own a superior good quality chef's knife, restrain yourself from all other kitchen purchases until you can justify a good one hundred$ or far more chef's knife. This makes the knives virtually indestructible, and the manage will under no circumstances separate from the blade. At least until they essentially have to use their knives Should you liked this post and you would like to acquire more info relating to Best Kitchen Knives Set Reviews (www.thebestkitchenknivesreviews.Com) i implore you to pay a visit to our own web site. .

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