(Cumulative distribution function) A dart is ung at a circular target of radius
ID: 3251458 • Letter: #
Question
(Cumulative distribution function) A dart is ung at a circular target of radius R = 3. We can think of the hitting point as the outcome of a random experiment.
a) For simplicity, we assume that the player is guaranteed to hit the target somewhere. The sample space of the experiment is: ? = (x,y) : x?2 + y2 < R?2 , where R = 3. The probability that the dart lands in some region A is proportional with area of A. Hence, we have:
P(A) = (Area(A)) / (?R2?) = (Area(A)) / (9?)
The scoring system is as follows. The target is partitioned by three concentric circles C1, C2, C3, centred at the origin with radii R1 = 1, R2 = 2 and R3 = R = 3. These circles divide the target into three annuli, which are described in Table 1:
Table 1: Formal denitions of the regions A1, A2, A3 and the probabilities assigned to them.
We suppose that the player scores an amount k if and only if the dart hits Ak, where k ? {1,2,3}. Assuming that the random variable X is the resulting score, then we have for k ? {1,2,3} that X = k if and only if the dart hits Ak. Write down the cumulative distribution function of X.
Explanation / Answer
The cumulative probability is as given is the table above.
The corresponding probabilities are according to the area of the region.
X Score ProbP(X=x) Cumulative prob(P(X<=x) 1 1 1/9 1/9 2 2 3/9 4/9 3 3 5/9 1Related Questions
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