Probability And Expected Value Linguee Apps
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In the long run, you won't lose any money, but you won't win any. Don't expect to see a game with these numbers at your local carnival.
If in the long run, you won't lose any money, then the carnival won't make any. Now turn to the casino.
In the same way as before we can calculate the expected value of games of chance such as roulette. In the U. Half of the are red, half are black.
Both 0 and 00 are green. A ball randomly lands in one of the slots, and bets are placed on where the ball will land. One of the simplest bets is to wager on red.
If the ball lands on a black or green space in the wheel, then you win nothing. What is the expected value on a bet such as this? Here the house has a slight edge as with all casino games.
As another example, consider a lottery. The probable rate of return of both the securities security P and Q are as given below.
Based on the given information, help Ben to decide which security is expected to give him higher returns. In this case, the expected value is the expected return of each security.
Let us take another example where John is to assess the feasibility of two upcoming development projects Project X and Y and choose the most favorable one.
Determine for John which project is expected to have a higher value on completion. It is important to understand for an analyst to understand the concept of expected value as it is used by most investors to anticipate the long-run return of different financial assets.
The expected value is commonly used to indicate the anticipated value of an investment in the future. On the basis of the probabilities of possible scenarios, the analyst can figure out the expected value of the probable values.
The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that, sometimes, it is difficult or impossible to use other approaches.
The average of the results obtained from a large number of trials may fail to converge in some cases. The median is zero, but the expected value does not exist, and indeed the average of n such variables has the same distribution as one such variable.
It does not converge in probability toward zero or any other value as n goes to infinity. The Italian mathematician Gerolamo Cardano — stated without proof that the accuracies of empirical statistics tend to improve with the number of trials.
He named this his "Golden Theorem" but it became generally known as " Bernoulli's Theorem ". This should not be confused with Bernoulli's principle , named after Jacob Bernoulli's nephew Daniel Bernoulli.
In , S. Poisson further described it under the name " la loi des grands nombres " "the law of large numbers".
After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev ,  Markov , Borel , Cantelli and Kolmogorov and Khinchin.
Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true.
One is called the "weak" law and the other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.
There are two different versions of the law of large numbers that are described below. They are called the strong law of large numbers and the weak law of large numbers.
Lebesgue integrability of X j means that the expected value E X j exists according to Lebesgue integration and is finite.
It does not mean that the associated probability measure is absolutely continuous with respect to Lebesgue measure. Large or infinite variance will make the convergence slower, but the LLN holds anyway.
This assumption is often used because it makes the proofs easier and shorter. Mutual independence of the random variables can be replaced by pairwise independence in both versions of the law.
The difference between the strong and the weak version is concerned with the mode of convergence being asserted. For interpretation of these modes, see Convergence of random variables.
The weak law of large numbers also called Khinchin 's law states that the sample average converges in probability towards the expected value .
Interpreting this result, the weak law states that for any nonzero margin specified, no matter how small, with a sufficiently large sample there will be a very high probability that the average of the observations will be close to the expected value; that is, within the margin.
As mentioned earlier, the weak law applies in the case of i. For example, the variance may be different for each random variable in the series, keeping the expected value constant.
If the variances are bounded, then the law applies, as shown by Chebyshev as early as If the expected values change during the series, then we can simply apply the law to the average deviation from the respective expected values.
The law then states that this converges in probability to zero. In fact, Chebyshev's proof works so long as the variance of the average of the first n values goes to zero as n goes to infinity.
At each stage, the average will be normally distributed as the average of a set of normally distributed variables.
The strong law of large numbers states that the sample average converges almost surely to the expected value . What this means is that the probability that, as the number of trials n goes to infinity, the average of the observations converges to the expected value, is equal to one.
The proof is more complex than that of the weak law. Almost sure convergence is also called strong convergence of random variables.
This version is called the strong law because random variables which converge strongly almost surely are guaranteed to converge weakly in probability.
However the weak law is known to hold in certain conditions where the strong law does not hold and then the convergence is only weak in probability.
See Differences between the weak law and the strong law. The strong law of large numbers can itself be seen as a special case of the pointwise ergodic theorem.
The strong law applies to independent identically distributed random variables having an expected value like the weak law.
This was proved by Kolmogorov in It can also apply in other cases. Kolmogorov also showed, in , that if the variables are independent and identically distributed, then for the average to converge almost surely on something this can be considered another statement of the strong law , it is necessary that they have an expected value and then of course the average will converge almost surely on that.
This statement is known as Kolmogorov's strong law , see e.