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Math Help - Probability pmf

  1. #1
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    Probability pmf

    Let X be the number of accidents in a factory per week. , It has the pmf

    f(x) = 1 / (x+1)(x+2) ; x=0,1,2,...

    Find the conditional probability of x > or = 4, given that x >or = 1
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  2. #2
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    Quote Originally Posted by amor_vincit_omnia View Post
    Let X be the number of accidents in a factory per week. , It has the pmf

    f(x) = 1 / (x+1)(x+2) ; x=0,1,2,...

    Find the conditional probability of x > or = 4, given that x >or = 1
    \Pr(X \geq 4 | X \geq 1) = \frac{\Pr(X \geq 4) ~ \text{and} ~ \Pr(X \geq 1)}{\Pr(X \geq 1)} = \frac{\Pr(X \geq 4)}{\Pr(X \geq 1)}.

    Do you know how to get these two probabilities from the pmf?
    Last edited by mr fantastic; June 17th 2008 at 01:28 AM.
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    Quote Originally Posted by mr fantastic View Post
    \Pr(X \geq 4 | X \geq 1) = \frac{\Pr(X \geq 4) ~ \text{and} ~ \Pr(X \geq 1)}{\Pr(X \geq 1)} = \frac{\Pr(X \geq 4)}{\Pr(X \geq 1)}.

    Do you know how to get these two probabilities from the pmf?
    f(x) = \frac{1}{(x+1)(x+2)} = \frac{1}{x+1} - \frac{1}{x+2}.

    Therefore \Pr(x \geq 4) = \sum_{x=4}^{\infty} \left( \frac{1}{x+1} - \frac{1}{x+2} \right) = \left( \frac{1}{5} - \frac{1}{6} \right) + \left( \frac{1}{6} - \frac{1}{7} \right) + ...... = \frac{1}{5}.

    (This is an example of a telescoping series)

    \Pr(x \geq 1) is got in a similar way ....
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  4. #4
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    Quote Originally Posted by mr fantastic View Post
    \Pr(x \geq 1) is got in a similar way ....
    Also, note that \Pr(x \geq 1) = 1 - \Pr(x < 1) \Rightarrow 1 - \Pr(x = 0)
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