Probability & Statistics/Probability

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Question
Hi,
Thankyou for your time & would appreciate an answer to a statement in a book I am reading regarding trading as I do not know how the answer is worked out. Perhaps you could explain the workings for me which would be greatly appreciated. How does the author arrive at a 65% chance? I have no problems with arriving at the average or standard deviation.

Let’s say your expectancy
is 0.33R, but your standard
deviation is 3R. What this means
is that even though your average
gain after 20 trades should total
about 6.6R, you only have about
a 65% chance of being profitable
after 20 trades because of the
huge variability.

Thankingyou in anticipation

Answer
The statement is explaining how mean and standard deviation are both important factors in determining the probability that one sees a particular type of outcome (in this case, a profit).


If the expectation (mean) is 0.33R, this just means that the average outcome is 0.33R.

Having a positive mean does not guarantee that you are likely to make a profit. For example, let's say you choose one of six numbers at random, and the possible outcomes are -1, -2, -3, -5, -7, or +20 outcome. Although the average is (20-1-2-3-5-7)/6 = 0.33R, you only have a 1/6 chance of picking a positive number.

However, under a normal distribution , a positive mean will guarantee a likelihood (better than 50% at least) that the outcome is positive, because half of all outcomes are higher than the mean (in the case you have, another 15% is also between 0 and the mean for a total of 65%).

For more on the normal distribution, see Wikipedia.

In a normal distribution there will still be some chance that the outcome is negative, and if the mean is close to zero (compared to the standard deviation), that chance will be significant.

For the expectancy (mean) equal to μ and standard deviation σ, the probability that you are above some threshold T can be computed. For example, the probability of being higher than T=μ is exactly 1/2. The probability of being higher than T=μ-σ is about 84%. The probability of being higher than T=μ-3σ is over 99%.

So if μ = 0.33R and σ = 0.03, it is a virtual certainty that the outcome is positive (about 97%).

If μ = 0.33R and σ = 0.3, then it is somewhat likely to be positive (73%).

And in the case given, where σ=3, it is about 57% likely to be positive.

For more about this, see Wikipedia.

Now, that's only true for a single transaction -- not twenty of them.

It's important that if you have multiple normal distributions, and they add up, then the μ and σ values add up. For two, it might be:

X1 is distributed by μ1 and σ1

X2 is distributed by μ2 and σ2

X3 is distributed by μ and σ

⇒ μ = μ1 + μ2, σ = σ1 + σ2

So for twenty identical trades, the new values of μ and σ are:

μ = 20*0.33 = 6.6

σ = 20*3 = 60

Notice that this means σ is only changed by √(20) factor, not 20.


So you can then use calculus to compute an integral from x=0 to x=∞ of 1/(σ√(2π) ∫ e^( (x-μ)^2 / 2σ ) dx with the values of μ and σ above to get 80.2. This is actually way higher than 65%, and I think your book may have a typo or error of some type.

For more on computing such things, see Wikipedia.


So, what is interesting to observe is this: Say you have n trades (n=1 and n=20 we have done, and seen 57% and 80% as probability of a positive outcome).

Well, if you have n=100 trades, it should be even higher than 80% right? The more trades you do, the closer you get to the average -- meaning you are less and less likely to dip below zero. This is because the mean is multiplied by n (which was 20 above), while the variance σ is only multiplied by √(n).

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Clyde Oliver

Expertise

I can answer all questions up to, and including, graduate level mathematics. I do not have expertise in statistics (I can answer questions about the mathematical foundations of statistics). I am very much proficient in probability. I am not inclined to answer questions that appear to be homework, nor questions that are not meaningful or advanced in any way.

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I am a PhD educated mathematician working in research at a major university.

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AMS

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Various research journals of mathematics. Various talks & presentations (some short, some long), about either interesting classical material or about research work.

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BA mathematics & physics, PhD mathematics from a top 20 US school.

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Various honors related to grades, various fellowships & scholarships, awards for contributions to mathematics and education at my schools, etc.

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In the past, and as my career progresses, I have worked and continue to work as an educator and mentor to students of varying age levels, skill levels, and educational levels.

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