Representative Matsui of Nippon Ishin no Kai answering an interview with a TV station at the ballot counting center = 10:10 pm on the 53th, a hotel in Osaka city

### Restoration, increase of seats also misses priority ward Matsui representative "insufficient legs"

** ***If you write the contents roughly*

It is a perception that expectations have risen a little, especially in Osaka and Kansai. "** **

The Nippon Ishin no Kai aims to double the number of seats in the six re-elections, and aims to receive criticisms against the Kishida administration.** → Continue reading**

**Kyodo News**

We will deliver the latest news in Japan and abroad in a precise and prompt manner.

### Wikipedia related words

If there is no explanation, there is no corresponding item on Wikipedia.

# Expected value

Probability theoryAtRandom variable Of**Expected value**(Kitaichi,British: expected value expected value) Means all values of random variablesprobabilityWeightedweighted average.Probability distributionWhen defining for, it is often called "average".

For example,gamblingIn some cases, it may be the "expected" amount of prize money that can receive the expected value.However, the probability of a random variable value that takes an expected value is not always the maximum, and the random variable value does not always take an expected value.but,Independent and identical distributionIf,Sample meanIs known to converge to the expected value (Law of large numbers).

Expected value is meaningfulElementary eventThis is the case when the probabilities of are even to some extent.If there is an elementary event with a very extreme probability, it does not fit into the concept of expected value.For example, in "insurance" and "lottery", the probability of getting a large amount of insurance money and winning money is very small, so the expected value is negative, but with that, getting insurance or purchasing the lottery is just a loss. It is not appropriate to judge that.

## Definition

### Discrete random variable

Probability space (Ω, *F*, *P*) AtRandom variable X At mostCountableIndividual *x*_{1}, *x*_{2}, ... When taking (Discrete random variable),X The expected value of

Defined in.

### Continuous random variable

Probability space AtRandom variable X When is a continuous value such as a real number (uncountable infinity) (Continuous random variable), Integrable random variable X The expected value of

Defined in.However, random variables X IntegrableIs

Is to meet and thisintegralIs abstractLebesgue integral.

Event Against

And write the range to take the expected value A Limit to.here 1_{A} TheIndicator function.

### Japanese Industrial Standard

Japanese Industrial Standard In (JIS), the value x_{i} AppearsprobabilityThe *p _{i}* = Pr {

*X*=

*x*} ToDiscrete probability distributionExpected value forProbability density function

_{i}*f*(

*x*) haveContinuous probability distributionDefines the expected value of.When the measurement is performed many times and the average of the measured values is calculated, the value is close to the expected value.function

*g*(

*X*) Expected value of

*E*[

*g*(

*X*)] Is defined in the same way.Also, the expected value of the conditional distributionConditional expectation,

*X*.

*Y*OfJoint distributionRegarding the conditions

*Y*=

*y*Under X Conditional expectation of y To be a function of, a random variable X Expected value of X Introducing the mother average of

^{[1]}.

## nature

Expected value isSum,integralSince it is defined by, it has all the properties of summation and integration.Less than,*X*, *Y* TheRandom variable,*a*, *b* Is a scalar.

- Linearity
- Monotonic
- Jensen's inequality : Convex function φ Against
- Chebyshev's inequality : (0, ∞) Positive value defined aboveMonotonically increasing function φ And any positive number ε Against

In addition, twoIntegrableRandom variables X と Y independentIn the case of,

Is established.

## Calculation method

Since the expected value of a continuous random variable is defined by the Lebesgue integral, it is common to perform variable transformation of the integral and integrate with the distribution of the random variable when calculating.Random variable X OfdistributedThe P_{X} Then any measurable function f Against

And then P_{X} Probability density function p When you have

ByLebesgue measureYou will be able to calculate with.

## An example

### Expected value of dice roll

Hexahedron OfDiceShake once.However, the probability of getting out is all 1/6 And.Eyes to come out X The expected value of

### Expected value of prize money

Consider the following game.

- If you pay 100 yen, you can roll one 6-sided dice once.
- The next amount depending on the roll of the dice X You can get a yen.

Eyes that came out | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

X | 20 | 50 | 100 | 100 | 150 | 150 |

At this time, the amount of money you can get X The expected value of

The participation fee is less than 100 yen.From this, it can be said that as the number of trials increases, the average loss of this game is 1 yen, and the more times the number of trials increases, the more the loss (Law of large numbers).

## footnote

## References

- Yasuo Nishioka "Mathematics Tutorial Easy Talking Probability Statistics"Ohmsha, 2013.ISBN 9784274214073.
- Koji Fushimi "Probability theory and statistics"Kawade Shobo, 1942.ISBN 9784874720127 .
- Mathematical Society of Japan "Mathematical Dictionary"Iwanami Shoten, 2007.ISBN 9784000803090.
- JIS Z 8101-1 1999 Statistics-Terms and Symbols-Part 1: Probability and General Statistical Terms(Japanese Standards Association）

## Related item

- Probability theory
- St. Petersburg paradox --Example where the expected value cannot be obtained
- Law of large numbers