Update Readme.md. Add math

README.md

@@ -24,4 +24,25 @@ model-index:
   # **Reinforce** Agent playing **Pixelcopter-PLE-v0**
   This is a trained model of a **Reinforce** agent playing **Pixelcopter-PLE-v0** .
   To learn to use this model and train yours check Unit 4 of the Deep Reinforcement Learning Course: https://huggingface.co/deep-rl-course/unit4/introduction
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+
+
+  # Some math about 'Pixelcopter' training.
+The game is to fly in a passage and avoid blocks. Let we have trained our agent so that the probability to crash at block is _p_ (low enogh, I hope).
+The probability that the copter crashes exactly at _n_-th block is product of probabilities it doesn't crash at previous _(n-1)_ blocks and probability it crashes at current block:
+$$P = {p \cdot (1-p)^{n-1}}$$
+The mathematical expectation of number of  the block it crashes at is:
+$$<n> = \sum_{n=1}^\infty{n \cdot p \cdot (1-p)^{n-1}} = \frac{1}{p}$$
+The std is:
+$$std(n) = \sqrt{<n^2>-<n>^2}$$
+$$<n^2> = \sum_{n=1}^\infty{n^2 \cdot p \cdot (1-p)^{n-1}} = \frac{2-p}{p^2}$$
+$$std(n) = \sqrt{\frac{2-p}{p^2}-\left( \frac{1}{p} \right) ^2} = \frac{\sqrt{1-p}}{p}$$
+So difference is:
+$$<n> - std(n) = \frac{1 - \sqrt{1-p}}{p}$$
+As long as
+$$ 0 \le p \le 1 $$
+the following is true:
+$$\sqrt{1-p} \ge 1-p$$
+$$<n> - std(n) = \frac{1 - \sqrt{1-p}}{p} \le \frac{1 - (1-p)}{p} = 1$$
+The scores _s_ in 'Pixelcopter' are the number of blocks passed decreased by 5 (for crash). So the average is lower by 5 and the std is the same. No matter how small _p_ is, our 'least score' is:
+$$ (<n> - 5) - std(n) = <n> - std(n) - 5 \le - 4$$
+But as we use only 10 episodes to calculate statistics and episode duration is limited, we can still achieve the goal, better agent, more chances. But understanding this is disappointing