53. A shot on a target (x, y) is uniformly distributed within a target disk of r

Question

53. A shot on a target (x, y) is uniformly distributed within a target disk of radius r. That is, the density is f(x,y) = C (for some appropriate C) for 0 x2 + y2-r2 (a) Find the probability that the distance from the origin of this point is less than x, 0 s x sr (b) Let x=r/4 and suppose that 10 independent shots are taken at the target with impacts governed by the uniform distribution. Use a common distribution to write the formula for the probability that three or less shots are within x of the origin

Explanation / Answer

a) this is uniform distribution

P(distance < x) = area of circle with radius x / area of circle with radius r

= x^2 / r^2

b)

P(distance < r/4) = 1/16   {replace x = r/4 in a) part)

Y = number of shots which are within r/4

Y follows binomial distribution with parameter n = 10 , p = 1/16

P(Y <= 3 )

= 0.99764

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