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Math Help - Cracking the Math GRE Questions - more errors?

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    Cracking the Math GRE Questions - more errors?

    Again from Princeton Review's fourth edition, chapter 7 review:

    #38 Let X be a random variable on \mathbb{Z}^+ (positive integers) whose distribution function is F_x (t) = \frac 1 {3^t}. Suppose that Y is another random variable whose distribution function is F_Y (t)= \frac 1 {4^t}. What is the probability that at least one of the variables X and Y is greater than 2?

    (a) 5/6
    (b) 64/81
    (c) 1/2
    (d) 17/81
    (e) 1/6

    First off, the question seems poorly worded since I am not sure if Y is also a random variable on the positive integers. Furthermore, shouldn't their distribution functions be increasing? The answer in the book is (e). The explanation of the answer seems to treat the functions F_X and F_Y as though they are describing the probability that X\ge t and Y \ge t instead of, what I thought probability distribution functions described, P(X\le t) and P(Y \le t). Here is the explanation:

    The problem asks for the probability that at least one of the variables is greater than 2. That's the same as 1 minus the probability that both variables are less than or equal to 2. Since X and Y are independent, we can multiply the complements of their individual probabilities. So

    \begin{align*}P(\text{at least one} >2) &= 1-P(X\le 2)P(Y\le 2)\\  &= 1-\left(\frac {3^2-1}{3^2} \right )\left(\frac{4^2-1}{4^2} \right )\\ &= 1-\frac 5 6\\ &= \frac 1 6\end{align*}


    I understand the logic of the first part, but why is it that they seem to say that P(X \le 2)=1-\frac 1 {3^2}? It seems to me that it would just be \frac 1 {3^2}, but then again those probability distribution functions don't make sense since they are decreasing...
    Last edited by process91; November 6th 2011 at 07:24 PM.
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    Re: Cracking the Math GRE Questions - more errors?

    Hello,

    I agree there is a 'problem' in what they call 'distribution function'. But as you say, cdf should be an increasing function, and here it's decreasing. So one can think that it is instead the survival function (which is 1-cdf). Not well worded.
    Yes, we need to know if F_Y is not 0 for t=2.
    Additionally, they don't clearly state if the rv's are independent...

    Well, several weird things in this problem...
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    Re: Cracking the Math GRE Questions - more errors?

    Thank you very much!
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