Calculate the probability of reversal
Use Pascal's triangle. You add up all the values in each row. That's 100%. Then take any point with its value and divide by its value. That's the probability.
Interestingly, I figured out the Pascal triangle on my own, I didn't even know it existed or what it was called). But doing it manually is not realistic, because if you take only 10 steps, you get 252 combinations in zero, which is a hell of a formula. Of course I can make the computer calculate it all, but maybe there is a more elegant way?
Maybe I got it wrong, I'll try it like you wrote.Use Pascal's triangle. You need to add up all the values in each row. This is 100%. Then take any point with its value and divide by the resulting value. That's the probability.
No, I already have the probability in percentages, I need to calculate what the probability of reversing in each step should be to get this distribution
No, I already have the probability in percentages, I need to calculate what the probability of reversal at each step should be to get this distribution
Is the starting point 17.9% (top of the normal distribution) or not? And I probably jumped the triangle, because there is no movement inside the triangle, it's all along the edges.
Is the starting point 17.9% (top of the normal distribution) or not? And about the triangle, I probably jumped the gun, as there is no movement inside the triangle, all along the edges.
Yes, in the example the probability of getting to the starting point (from which you left) is 17.9%, i.e. the top of the distribution. It turns out that with a probability of 17.9%, in 10 steps it will return to where it came from.
Well, then I was right about the triangle. Because you only need calculations for the faces, for each point on the face you take its coefficient. For example, for points 16.06% and 16.01%, the coefficient is 0.5, because the second line consists of two units. Then, for 16.01%, the probability is (17.9 + 0.5 * 16.01) / 2 = 12.9525%, and for 16.06%: (17.9 + 0.5 * 16.06) / 2 = 12.965%
For points 11.89% and 11.9%, a factor of 0.25 applies, as the numbers in the third row are: 1, 2, 1. Then for 11.89%: (12.9525 + 0.25 * 11.89) / 2 = 7.9625%, and for 11.9%: (12.965 + 0.25 * 11.9) / 2 = 7.97%.
That is, for each new point, the probability of the previous step is taken, its point value is added, multiplied by the coefficient for the given series, and divided by 2. It is solved by the usual loop on the indexes of the triangle series, no need to try to cram everything into one formula.
Well, then I was right about the triangle. Because you only need calculations for the faces, for each point on the face you take its coefficient. For example, for points 16.06% and 16.01%, the coefficient is 0.5, because the second line consists of two units. Then, for 16.01%, the probability is (17.9 + 0.5 * 16.01) / 2 = 12.9525%, and for 16.06%: (17.9 + 0.5 * 16.06) / 2 = 12.965%
For points 11.89% and 11.9%, a factor of 0.25 applies, as the numbers in the third row are: 1, 2, 1. Then for 11.89%: (12.9525 + 0.25 * 11.89) / 2 = 7.9625%, and for 11.9%: (12.965 + 0.25 * 11.9) / 2 = 7.97%.
That is, for each new point, the probability of the previous step is taken, its point value is added, multiplied by the coefficient for the given series, and divided by 2. Solved by the usual loop on the row indexes of the triangle, no need to try to cram everything into one formula.
Here is an example in the picture. There are 2 cases. In the top one the probability of reversal at each step is 50%, i.e. the process has no memory, then you get the probability density distribution as drawn. It is very easy to calculate the probability of reversal for the extreme values only (12.5/100)^(1/3)=0.5. That is, the probability of reversal for the extreme value is easily calculated but for 37.5 we do not know how to calculate the probability of reversal.
The figure below is more complicated as the process already has memory where the probability that the next step will be in the same direction as the previous one is 0.6 and the probability of reversal is 0.4. The probability density function is therefore different from the previous case. Hence the question how to calculate the probability of reversal using only the probability density function.
Here too, we can take the extreme value (18/100)^(1/3)=0.56 which is the average probability of reversal as it was 0.5 at the first step.
But how can we find probability of reversal for values of 32?
Maybe I'm thinking in the wrong way and there is a way that is significantly different from what I've shown? That is, I need to calculate from the shape of the distribution what the average probability of reversal (or continuation) resulted in that particular shape of the distribution.
At first sight, the usual problem from the field of Markov chains is the evolution of the initial distribution over time. Some complication is due to the fact that the chain is of the second order (the probability of price at moment n depends not only on the price at moment n-1, but at moment n-2 as well)
The calculation has to be done numerically. Elegantly (analytically) one could only calculate the stationary distribution, but here it is obviously not defined.
in a normal distribution there is no memory and the probability of each next step being directed =50%.
There is no memory in any distribution. The probability of continuation/reversal is not determined by the kind of distribution, but by the correlation of the increments (in the most general case).
From the type of distribution of the increments you can determine the other - the probability of reaching a certain level in a certain time (if I understand correctly, I'm not a mathematician).
Such problems are found in option calculations, google it.
But you seem to want to use a value distribution - I can't say anything here.
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Who is good at maths, please help me solve this problem, I can't figure out how to do it.
We have a probability density plot for normal distribution, in normal distribution there is no memory and probability of direction of each next step =50%.
Suppose we have a person who takes 10 steps, he can step to the right or to the left, each next step is independent of the previous one and the probability of going left or right is 50%. Then we can build a table of probability densities and estimate with what probability he will move away from the starting point in 10 steps. The 6th column shows the probability in %. The table shows that with probability 0.0977% he will move to the right from the starting point for 10 steps or with probability 4.39% he will move 6 steps for 10 steps.
It's simple, the probability of reversal is always 50%, but if the probability of reversal is different than 50%, the probability density graph will be different.
And hence the question, how with only the probability density graph, to calculate the probability of reversal at each step.
Let's say we have this probability density graph
Here on x-axis you can see how many steps the person went from the starting point, from -10 (to the left) to +10 (to the right) and signed with what probability he did it in %. How do I find the probability of each step being reversed?