(Cumulative distribution function) A dart is ?ung at a circular target of radius
ID: 3236749 • 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:
Table 1: Formal de?nitions of the regions A1, A2, A?3 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.
(***I could not find a way to type the bar symbol over the letter H to make it the complement of H, so the last two Hbar? variables are meant to represent the complements of H)
Use the identity above in order to write down the cumulative distribution function of Y .
Definition Probability Q=(x,y) : r2 + y2Explanation / Answer
The cumulative probability distribution is as follows:
F(1) = 1/9
F(2) = 1/9+3/9 = 4/9
F(3) = 4/9+5/9 = 1
The resulting score can be if the target is hit with certain probability or the target is not hit with certain probability
If it is known that target is hit, then prob. = P(y<=y|H).p(H), this means that it is known that H occurs and we know p(H), and having H occured the score prob is P(Y<=y|H).
SImilarly when the target is missed, the prob. is P(Y<=y|H bar).P(H bar)
The player can either hit the target or miss the target, so either event H or H bar can happen
Hence P(Y<=y) = P(y<=y|H).p(H)+ P(Y<=y|H bar).P(H bar)
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