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Area under the curve - probability

  1. Jun 26, 2010 #1
    It's something that I know but I've never been able to figure out 'why?' Why is does the area underneath the normal distribution (or any distribution) represent probability?

    Thank you.

  2. jcsd
  3. Jun 26, 2010 #2


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    The short answer to your question is the definition:

    A nonnegative function f(x) is said to be the density function for the continuous random variable X if for all real numbers a < b:

    [tex]P(a\le X \le b) = \int_a^b f(x)\, dx[/tex]

    From calculus we know the integral on the right represents the area under f(x) between a and b. Since the cumulative distribution function is given by

    [tex]P(X \le x) =\int_{-\infty}^{x} f(t)\, dt[/tex]

    it follows that

    [tex]\int_{-\infty}^{\infty} f(x)\, dx = 1[/tex]

    Putting these together shows why the area under the curve represents probability. Does that explanation help or did you already know that?
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