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PDF of a continuous random variable

  1. Nov 19, 2014 #1
    1. The problem statement, all variables and given/known data
    Let X denote a continous random variable with probability density function f(x) = kx3/15 for 1≤X≤2. Determine the value of the constant k.

    2. Relevant equations
    I'm not sure if this is right but I think ∫kx3/15 dx=1 with the parameters being between 2 and 1,

    3. The attempt at a solution
    So I did what I i showed in the relevant equations section. I got k = 4. When I plug 2 into the equation for X, my probability is greater than 1 so I know this isn't right. I'm not sure what else to do.
     
  2. jcsd
  3. Nov 19, 2014 #2

    Orodruin

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    By this do you mean the PDF f(x) > 1? There is nothing wrong with this as f(x) is the probability density and not a probability in itself.
     
  4. Nov 19, 2014 #3

    Ray Vickson

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    As Orodruin has indicated, f(x) can be greater than 1; the probability that ##X## lies in the interval ##(x,x + \Delta x)## is (for small ##\Delta x > 0##) given by ##P(x,x+\Delta x) = f(x) \cdot \Delta x + o(\Delta x)##. Here, the notation ##o(h)## means terms of higher order in small ##h> 0##; that is ##o(h)/h \to 0## as ##h \to 0##. In other words, for small ##\Delta x > 0## the probability is nearly proportional to ##\Delta x##, with coefficient ##f(x)##. Even if we have, say ##f(x) = 10,## the probability would be ##10 \times 0.0001 = 1/1000 << 1## if ##\Delta x = 0.001##.
     
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