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Circular Unifrom Dist'

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  • Circular Unifrom Dist'

    A dart is thrown at a dashboard with radius of 7 cm. The point that the darrt hits is uniformly distributed on the circular dashboard. Find the expected distance of that point from the center of the dashboard.

    Let R = radius

    P(R > r ) = P(X^2 + Y^2 > r^2)
    = [ 49(pi) - (pi)r^2 ] / 49(pi)

    E[R] = int[ P(R > r ) ]
    = 14/3

    My question is: how did he get [ 49(pi) - (pi)r^2 ] / 49(pi)

  • #2
    consider the entire circle. In this solution he wants to find Survival function (Probability that the the dart is thrown OUTSIDE of a distance r..... Once you have the S[x], You integrate to get E[x] yes?

    Well 49 pi is your circles area.... and you'd like to know the probability of hitting a length of r or greater. Since it's uniform, you just deduct the area of NOT hitting r (distance from 0 to r) and you'd have the area of hitting the area you'd want yes?

    It's just entire circle minus a smaller circle that you don't want out of the entire circle...

    It's kinda hard to illustrate this to you without drawing circles! hehe... but I hope I helped you out!