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For integers 0 to 100, I am trying to write a series of constraints that define a bell curve shape centered around 50 and mass =0.5.

Informally, I used my own logic to come up with a set of constraints to make this happen:

  • Suppose I have variables p0, p1, p2, p3 ....p100.
  • I know that p0+ p1+p2+p3+....+p100 = 1
  • I know that p50 = 0.5
  • I know that (p49 = p51) > (p48 = p52) > (p47 = p53), etc.
  • I know that any given p0, p1,p2,p3..p100 must be between 0 and 1.
  • And pi - pj = pj - pk where i >j >k and i-j = 1 and j-k = 1

My Problem: I am trying to solve for the values of p1 to p100.

I think that some form of Linear/Integer Programming can be used to solve this problem, but I am not sure how to begin.

I wrote the all the constraints in mathematical form:

$$ \text{Constraint 1:}\ \quad \sum_{i=0}^{100} p_i = 1 $$

$$\text{Constraint 2:}\ \quad p_{50} = 0.5$$

$$\text{Constraint 3:}\quad p_{50-i} = p_{50+i} > p_{50-2i} = p_{50+2i}\ \quad\forall i \in \{0, 1, 2, \ldots, 25\}]$$

$$ \text{Constraint 4:}\quad \quad 0 < p_i \leq 0.5 \quad \forall i \in \{0, 1, 2, \ldots, 49,51, \ldots, 100\} $$

$$ \text{Constraint 5:}\quad p_i - p_j = p_j - p_k \quad \forall i, j, k\ \quad \text{when:} \quad i >j > k \quad \text{and} \quad i-j = 1 \quad \text{and} \quad j-k=1 $$

But from here, I am not sure how to set up a system of equations to solve these relationships. I also do not know if I have correctly defined the constraints for this problem.

Can someone please show me how to do this correctly?

Thanks!

stats_noob
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    Something is [very] wrong with constraint 5. What behavior are you trying to capture with that constraint? – Brian Moehring Nov 11 '23 at 07:10
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    Constraint $3$ doesn't match your earlier definition. Also, this seems clearly impossible: (only considering $p_1$ to $p_{50}$) you're trying to get $50$ equally-spaced numbers between $0$ and $0.5$ to sum to $1$, with the maximum number being $0.5$ itself. Finally, what has this to do with the bell curve? – Benjamin Wang Nov 11 '23 at 07:11
  • The upper bound of constraint 4 is a consequence of constraint 2 and a (properly stated) constraint 3. As @BenjaminWang pointed out, you didn’t express it correctly. Also constraint 5, as written, is capturing something that isn’t true of a bell curve. – Joe Nov 11 '23 at 07:12
  • @ Brian: thank you for your reply! I also think constraint 5 might be wrong. Can you please show me how I can fix this? Thanks! – stats_noob Nov 11 '23 at 15:05
  • @ Benjamin: thank you for your reply! I thought that this constraint would capture the symetrical properties of a bell curve. – stats_noob Nov 11 '23 at 15:08
  • @ Joe: thank you for your reply! Can you please show ms how to write this correctly? – stats_noob Nov 11 '23 at 15:10
  • @stats_noob based on what you wrote above, it seems like you want the following two constraints two replace constraint 3. First, you want to impose the symmetry that for all $i = 1, \cdots, 5$, $p_{50 - i} = p_{50-i}$. Second, you want to impose that your probability mass is montonic deceasing as you move away from $p_50$. You can leverage the previous equality and just say for all $50 \leq i \leq 99$, we have $p_{i} > p_{i + 1}$ – Joe Nov 11 '23 at 18:51
  • @stats_noob Unless you want to answer the question I asked in the first comment, my advice for constraint 5 is to delete it. (By the way, putting an @ followed by a space and then the name won't notify the person about your message. Don't use spaces when @ ing someone) – Brian Moehring Nov 11 '23 at 20:50
  • Do you want to make a discrete version of a literal Bell/Normal/Gaussian curve? Or do you simply want something that has a probability distribution that is symmetric with nice round and decreasing curves (that can be made discrete)? – DotCounter Nov 14 '23 at 23:24

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