I have a convex polytope obtained by the intersection of set of halfspaces. Is there any way to find the probability that a random point on real space lies inside the convex polytope. Thanks in advance.
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2What is "a random point in the real space"? – joriki May 21 '20 at 21:10
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I apologize, it's any point in the 2-dimensional real space. – Mounica Devaguptapu May 21 '20 at 21:15
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Are all points equally likely to be drawn? – sudeep5221 May 21 '20 at 21:23
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Yes they are equally likely. – Mounica Devaguptapu May 21 '20 at 21:39
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1If the polytope has finite area, any point has probability $0$ of being inside. – herb steinberg May 21 '20 at 22:01
2 Answers
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There is no uniform distribution on $\mathbb R^2$; see Uniform distribution on $\mathbb Z$ or $\mathbb R$.
joriki
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You might want to consider the following slightly changed setup: What is the probability of a random point from some encasing hypercube (of finite size!) to also fall inside your convex polytope?
Then it becomes quite easy. Simply calculate the hypervolume $V$ of your convex polytope, calculate the hypervolume $V_0$ of the hypercube, and get the searched for probability $P$ by $$P=\frac{V}{V_0}$$ --- rk
Dr. Richard Klitzing
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