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Homework Help: 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.

    M
     
  2. jcsd
  3. Jun 26, 2010 #2

    LCKurtz

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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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