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Math Help - Cracking the Math GRE Questions - errors?

  1. #1
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    Cracking the Math GRE Questions - errors?

    This is from the fourth edition Princeton Review book, Chapter 7 review:

    #32 - If f(x) = \left\{ \begin{array}{lr} \frac x 2 + c \qquad \text{for } 0\le t \le 8 \\ 0 \qquad \text {otherwise} \end{array} \right., for what value of c is f(x) the probability density function of a random variable X?

    I know that the probability density function f(x) has to satisfy \int_{-\infty}^{\infty}f(x)dx=1, which for this function is just \int_0^8 \frac x 2 + c dx = 8^2 / 4 + 8c=1, therefore c=-\frac 7 8, but that is not an option. The options are 4/3, 2/3, 0, -2/3, and -4/3. The book claims answer -4/3 is correct, but their explanation also seems way off (they have that the integral above is equal to 8^2/4+6c=1, which still doesn't mean c should be -4/3).

    Can someone please either verify that I am correct or let me know where I went wrong?
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  2. #2
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    Re: Cracking the Math GRE Questions - errors?

    Quote Originally Posted by process91 View Post
    This is from the fourth edition Princeton Review book, Chapter 7 review:

    #32 - If f(x) = \left\{ \begin{array}{lr} \frac x 2 + c \qquad \text{for } 0\le t \le 8 \\ 0 \qquad \text {otherwise} \end{array} \right., for what value of c is f(x) the probability density function of a random variable X?

    I know that the probability density function f(x) has to satisfy \int_{-\infty}^{\infty}f(x)dx=1, which for this function is just \int_0^8 \frac x 2 + c dx = 8^2 / 4 + 8c=1, therefore c=-\frac 7 8, but that is not an option. The options are 4/3, 2/3, 0, -2/3, and -4/3. The book claims answer -4/3 is correct, but their explanation also seems way off (they have that the integral above is equal to 8^2/4+6c=1, which still doesn't mean c should be -4/3).

    Can someone please either verify that I am correct or let me know where I went wrong?
    You are correct with your logic but since \frac{8^2}{4}=\frac{64}{4}=16 we actually get that c=\frac{-15}{8}
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  3. #3
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    Re: Cracking the Math GRE Questions - errors?

    Whoops, yes - thank you.
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