# Robot and Probability

Robot drags parts from the conveyor to the box, and the number of parts it drags is randomly selected from $1$ to $k$. He drags the parts until there are $\ge n$ parts in the box. What is the expected value $\Epsilon \left[ X \right]$ 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 $m \le n$, where $1 \le k_i \le k$ is number of the dragged parts in $i$ action: $\begin{cases} { k }_{ 1 }+{ k }_{ 2 }+\dots +{ k }_{ m }\ge n \\ { k }_{ 1 }+{ k }_{ 2 }+\dots +{ k }_{ m-1 } For $m = 1$ system is equivalent to inequality: $k_1 \ge n;$Then use the formula of expected value, where $N_{i}$ is solutions number of system above for $m = i$: $\Epsilon \left[ X \right] =\sum _{ i=1 }^{ n }{ { x }_{ i }{ p }_{ i } } =\sum _{ i=1 }^{ n }{ i\cdot \frac { { N }_{ i } }{ \sum _{ j=1 }^{ n }{ { N }_{ j } } } ; }$ But i don't know how to solve system of inequalities. Please, help to understand. And It's interesting to see your ideas!

- 9 months, 3 weeks ago

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

- 9 months, 3 weeks ago

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

- 9 months, 2 weeks ago

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

- 9 months ago

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

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

Now you must think of a possible way to get the value of $N_i$ and determine $T_i$.

- 9 months ago

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?)

- 1 month, 1 week ago

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

- 8 months, 2 weeks ago

This isn't the right forum.

- 8 months, 1 week ago