Robot and Probability

Robot drags parts from the conveyor to the box, and the number of parts it drags is randomly selected from 11 to kk. He drags the parts until there are n\ge n parts in the box. What is the expected value E[X]\Epsilon \left[ X \right] of such actions?

Note by Ilya Pavlyuchenko
9 months, 3 weeks ago

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First idea: find the number of system solutions for each natural mnm \le n, where 1kik1 \le k_i \le k is number of the dragged parts in ii action: {k1+k2++kmnk1+k2++km1<n;\begin{cases} { k }_{ 1 }+{ k }_{ 2 }+\dots +{ k }_{ m }\ge n \\ { k }_{ 1 }+{ k }_{ 2 }+\dots +{ k }_{ m-1 }<n \end{cases}; For m=1m = 1 system is equivalent to inequality: k1n;k_1 \ge n;Then use the formula of expected value, where NiN_{i} is solutions number of system above for m=im = i: E[X]=i=1nxipi=i=1niNij=1nNj;\Epsilon \left[ X \right] =\sum _{ i=1 }^{ n }{ { x }_{ i }{ p }_{ i } } =\sum _{ i=1 }^{ n }{ i\cdot \frac { { N }_{ i } }{ \sum _{ j=1 }^{ n }{ { N }_{ j } } } ; } But i don't know how to solve system of inequalities. Please, help to understand. And It's interesting to see your ideas!

Ilya Pavlyuchenko - 9 months, 3 weeks ago

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I think I would basically use your approach and use a computer to evaluate all the possibilities

Steven Chase - 9 months, 3 weeks ago

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Are you asking for the expected value of the number of actions, or the expected number of parts in the box? (What is XX?)

Patrick Corn - 9 months, 2 weeks ago

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I asking for the expected value of the number of actions. And interesting how to solve system of inequality above.

Ilya Pavlyuchenko - 9 months ago

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Instead of j=1nNj\sum_{j=1}^n N_j, we want the number of combinations of actions from ii actions. (We are doing: Actions we care about divided by possible actions, and summing over the number of possible actions).

Call this value TiT_i. So E[X]=i=1ni×NiTiE[X] = \sum_{i=1}^n i \times \frac{N_i}{T_i} .

Now you must think of a possible way to get the value of NiN_i and determine TiT_i.

Alex Burgess - 9 months ago

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Pretty interesting question. For me, math is just like games. I think I need to read more about this gold question and then I will solve the issue. Everyone has their own games, right?)

Abbey Luxton - 1 month, 1 week ago

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AJAP1 is a complete transmembrane protein with 411 amino acid residues and its structure includes a separable N-terminal signal peptide (residues 1-43), extracellular domain (residues 44-282), transmembrane domain (residues 283-303), and intracellular cytoplasmic domains (residues 304-411).

https://www.creative-biogene.com/genesearch/AJAP1.html

Wendy Wilson - 8 months, 2 weeks ago

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This isn't the right forum.

Alex Burgess - 8 months, 1 week ago

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