Hope: indicator of chance or average risk

The expectation is a numerical value making it possible to measure the degree of equity of a Game of chance. It is equal to the sum of the profits (and the losses) balanced by the Probabilité of the profit (or of the loss).

Example of the Caster: by playing a full number, you have 1 chance out of 37 (the numbers go from 0 to 36) of touching 35 times your setting initial. By misant 10 euros, your hope of profit is thus:

$-10\; +\; \backslash \; frac\; \{10\; \backslash \; times\; 35\}\; \{37\}\; =\; -0,54$ (the 10 euros of setting are spent with a probability equalizes to 1)

This score indicates that on average, you will lose 54 centimes with each part with the profit of the casino. When the hope is equal to 0, it is said that the play is equitable .

Expectation and rational choice

In certain cases, the indications of the expectation do not coincide with a rational choice. Let us imagine for example that you the following proposal is made: if you manage to make a double six with two dice, you gain a million euros, if not you lose 10  000 euros. It is probable that you will refuse to play. However the hope of this play is very favorable for you: the probability of drawing a double 6 is of 1/36; one thus obtains:

$\backslash \; frac\; \{1\; \backslash ,\; 000\; \backslash ,\; 000\}\; \{36\}\; -\; \backslash \; frac\; \{10\; \backslash ,\; 000\; \backslash \; times\; 35\}\; \{36\}\; =\; 18\; \backslash ,\; 055$

with each part you gain on average 18  000 euros.

The problem precisely holds on this “on average”: if the profits are extremely important, they only intervene relatively seldom, and to have a guarantee reasonable not to finish ruined, it is thus necessary to have sufficient money to take part in a great number of parts. If the settings are too important to allow a great number of parts, the criterion of the expectation is thus not suitable.

Incidence of the allowance for risk

These are the considerations of risk of ruin which led, starting from its “paradox of Saint Petersbourg”, the mathematician Daniel Bernoulli to introduce in 1738 the idea of Aversion to the risk which results in matching the expectation of a Allowance for risk for its application in the questions of choice.

Particular applications (economy, insurance, finance, plays)

• the concept of allowance for risk applied to the expectation was in economy at the origin of the concept of Utilité (and of utility known as “marginal”).

• the premiums of Assurance are of one higher costs to the expectation of loss of the subscriber to the contract. But it is this risk of strong loss in the event of rare event which encourages it to subscribe it.
• the expectation, like other probabilistic concepts, is used in calculations of evaluation in Finance, for example for the evaluation of company.
• the behavioral Finance approaches, inter alia, the emotional and cognitive aspects, which go beyond the simple allowance for risk, and which can interfere with the rational concept of expectation per hour of the choice.
• Just as one pays a premium for to avoid the risks with the insurances, one pays on the contrary a access to the risk in the games of chance (which pay always less than their expectation, since they must be financed)

Concept of utility probabilistic

Rather than to pass by a concept of premium, one can directly establish a function of Utilité, associating with any couple {profit, probability} a value. The expectation then constitutes simplest of the functions of utility, suitable in the case of a player having very large resources at least in the absence of infinite.

Emile Borel adopted this concept of utility to explain why a player having few resources rationally chooses to take a lottery ticket each week: the corresponding loss is indeed for him only quantitative, while the profit - if profit there is - will be qualitative, its whole life while being changed. A chance on a million to gain a million can thus be worth in this precise case good more than an euro.

Mathematical aspect

The expectation of a random variable is the equivalent in Probabilité of the average of a statistical series in Statistiques. It notes E (X) and is read hope of X .

The hope is calculated, like the Variance, as from the moments of a random variable.

Formulas

The hope is defined for the random variables with values in R (or C) in the following way:

• Case of a discrete variable:
• If X takes a number finished N of actual values: x₁, x₂,…, x_n with the probabilities p₁, p₂,…, p_n then
  $E\; (X)\; =\; \backslash \; sum\_\; \{i=1\}\; ^\; \{K\}\; p\_i\; \backslash ,\; x\_i$
• If X takes a countable number of actual values: x₀, x₁,…, x_i,…. with the probabilities p₀, p₁,…, p_i,…. then
  $E\; (Z)\; =\; \backslash \; sum\_\; \{I\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\}\; p\_i\; \backslash ,\; x\_i$ if the series converges absolutely.

(absolute convergence ensures that the division of the series does not depend on the manner of numbering the terms)

• Case of a variable with density of probability:

• If X then has for Densité of probability F
  $E\; (X)\; =\; \backslash \; int\_\; \{\backslash \; mathbb\; \{R\}\}\; X\; \backslash ,\; F\; (X)\; \backslash ,\; dx$ provided that this integral exists.
• Case of a measurable application on a space of probability
• If X is a measurable application of (Ω, B, p) in R, positive or P-measurable,
  $E\; (X)\; =\; \backslash \; int\_\; \{\backslash \; Omega\}\; X\; \backslash ,\; dP\; =\; \backslash \; int\_\; \{\backslash \; mathbb\; \{R\}\}\; X\; \backslash ,\; dP\_X$ (where $\backslash \; P\_X$ is the probability image).

Estimate

The Loi of the great numbers makes it possible to say that the empirical Moyenne of NR observations (NR large) of the random variable X is a good estimate of the hope of X.

Central character

One frequently regards the hope as the center of the random variable, i.e. the value around which disperse the others valeurs.
In particular, if X and 2a - X have even law of probability, i.e. if the law of probability is symmetrical compared to has, then E (X) = A.

But this point of view is not valid any more when the law is dissymmetrical. To convince itself some it is enough to study the case of a geometrical Loi, a particularly dissymmetrical law. If X represents the number of throws necessary to obtain figure 1 with a cubic die, it is shown that E (X) = 6 what wants to say that one needs on average 6 throws to obtain figure 1. However, the probability that 5 tests or less are enough is worth nearly 0,6 and the probability that 7 throws or more are necessary is of 0,33. The values of X are thus not distributed equitably on both sides a hope.

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