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Statistics - related pdfs

  1. Feb 11, 2010 #1
    1. The problem statement, all variables and given/known data

    I am given the lens maker's equation:

    1/u + 1/v = 1/f

    Then told that U and V are random variables based on this equation.

    U is uniformly distributed between 2f and 3f.

    The question is to prove the pdf of V is f/(v-f)^2 and find the cdf for V. Also - find the mean and mode of V.

    2. Relevant equations

    3. The attempt at a solution

    The PDF for U seems to be:

    f_u = U/f for U 2f < U < 3f (not they should be 'less than and equal etc.)
    f_u = 0 otherwise

    And the cumulative distribution seems to be


    0 for U < 2f
    U-2f/f for 2f < U < 3f
    1 for 3f < U

    Now my attempts to translate into V have failed. I tried the transformation u = fv/f-v but with no luck in getting any algebra that is meaningful.

    I realise that f is related to both u and v in a way that makes simple substitutions of probablity difficult to incorporate. Some help would be appreciated!
  2. jcsd
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