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Math Help - Continuous random variable and cdf

  1. #1
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    Continuous random variable and cdf

    Consider the dartboard shown in the picture. Suppose that the diameter of the innermost circle in the middle (which has the point value of 10) is 2 units, and the width of each of the bands 19 is 1 unit, so that the radius of the circle up to the outer edge of the white 1-band is 10 units. Suppose that a player throwing a dart at the board has a probability of 1 of hitting this circle of radius 10, but that he is equally likely to hit any point within this circle. This means that his probability of hitting any region within the circle is proportional to the area of that region.

    Let X denote the distance from the location of a randomly thrown dart to centre of the board. Write down the cumulative distribution function of X.

    Here is the diagram:

    Continuous random variable and cdf-dartboard.png
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  2. #2
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    Re: Continuous random variable and cdf

    Quote Originally Posted by sakuraxkisu View Post
    Consider the dartboard shown in the picture. Suppose that the diameter of the innermost circle in the middle (which has the point value of 10) is 2 units, and the width of each of the bands 19 is 1 unit, so that the radius of the circle up to the outer edge of the white 1-band is 10 units. Suppose that a player throwing a dart at the board has a probability of 1 of hitting this circle of radius 10, but that he is equally likely to hit any point within this circle. This means that his probability of hitting any region within the circle is proportional to the area of that region.

    Let X denote the distance from the location of a randomly thrown dart to centre of the board. Write down the cumulative distribution function of X.

    F(X \leqslant t) = \left\{ \begin{gathered}  0,\quad t < 0 \hfill \\  \frac{{t^2 }}{{100}},\quad 0 \leqslant t < 10 \hfill \\  1,\quad 10 \leqslant t \hfill \\ \end{gathered}  \right.
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