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Homework Help: How to solve for probability given density function

  1. Feb 4, 2012 #1
    1. The problem statement, all variables and given/known data
    The probability density function of a random variable y is:
    f(y) = 100ye[itex]^{-10y}[/itex], if y>0
    f(y) = 0 otherwise

    What is the probability that 45y <= 10?



    2. Relevant equations
    E(y) = ∫yf(y)dy
    Var(y) = ∫(y-E(y))f(y)dy



    3. The attempt at a solution
    I solved for the expected value of y and got E(y) = 0.2
    I then got Var(y) = 0.02
    I don't think this is a normal distribution because the PDF is not a normal
    PDF. I don't know how to solve for the probability of 45y <= 10

    Any help will be greatly appreciated!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 4, 2012 #2
    Not quite sure, but maybe you do ∫f(y) dy from 0 to 10/45?
     
  4. Feb 4, 2012 #3
    I thought about that, but it just didn't seem right, after all, if one of the y's is over 10/45, the other y's can be less than that to make up for it.
     
  5. Feb 4, 2012 #4

    Curious3141

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    Homework Helper

    You shouldn't be worrying about what sort of distribution this represents. You're already given a pdf. Compute the cdf (cumulative distribution function). If the pdf is [itex]f(y)[/itex], the cdf is [itex]\int_{-\infty}^y f(y)dy[/itex]. Remember that from -∞ to 0, f(y) = 0, so break it up into two integrals, one from -∞ to 0 (which vanishes), and the other from 0 to y.

    Do the integration by parts and work out the definite integral in terms of y. Then it's as simple as substituting y = 10/45 into that.

    The answer I get lies between 0.6 and 0.7, if you wish to check yours.
     
  6. Feb 4, 2012 #5
    Ahh yes, I see, thank you very much. I got the same answer.
    Thanks again!
     
  7. Feb 4, 2012 #6

    Curious3141

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    Homework Helper

    No problemo. :smile:
     
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