6. We say that a point is selected at random from the region R whenever the...

 

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6. We say that "a point is selected at random from the region R" whenever the probability of selecting a point from inside a sub-region ACR is simply Area (A)/Area(R). That is to say, we use the probability measure Area (A) Area(R) (This is basically a 2-dimensional analog of the uniform probability measure in one dimension) P(A) = = A point is selected at random from the interior of the unit disk R := {(x, y) : x² + y² ≤ 1}; let R denote the distance from this point to the origin. (a) What is SR, the state space of R? (b) Find an expression for FR(r), the cumulative distribution function (c.d.f.) of R. Hint: What does the sub-region A := {R: R ≤r} look like geometrically? (c) Use your answer from part (b) compute E(R). Do NOT first find the probability density function (p.d.f.); there is a formula that relates E(R) to Fr(r) alone. (d) Now, use your answer from part (b) to find the probability density function (p.d.f.) fr(r) of R. (e) Use your answer from part (d) to verify your answer from part (c). 6. We say that "a point is selected at random from the region R" whenever the probability of selecting a point from inside a sub-region ACR is simply Area (A)/Area(R). That is to say, we use the probability measure Area (A) Area(R) (This is basically a 2-dimensional analog of the uniform probability measure in one dimension) P(A) = = A point is selected at random from the interior of the unit disk R := {(x, y) : x² + y² ≤ 1}; let R denote the distance from this point to the origin. (a) What is SR, the state space of R? (b) Find an expression for FR(r), the cumulative distribution function (c.d.f.) of R. Hint: What does the sub-region A := {R: R ≤r} look like geometrically? (c) Use your answer from part (b) compute E(R). Do NOT first find the probability density function (p.d.f.); there is a formula that relates E(R) to Fr(r) alone. (d) Now, use your answer from part (b) to find the probability density function (p.d.f.) fr(r) of R. (e) Use your answer from part (d) to verify your answer from part (c).

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a SR r 0 r 1 b FRr PR r r 1 F... View the full answer