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If X ∼ Uniform(−1, 1) find the pdf of Y = |X|

  1. Nov 25, 2015 #1
    This question is killing me.

    I know the graph is non-monotonic so i have to split up finding F(Y) for -1<Y and Y<1 but then what do I do with the modulus? >.<

    Any help would be greatly appreciated! Thank you so much x
     
  2. jcsd
  3. Nov 25, 2015 #2

    andrewkirk

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    You know the pdf of X, which is as simple as a pdf can be.
    You have correctly assessed that the next step is to work out the cdf ##F_Y## of Y. THat is, for a given ##y\in[0,1]##, we need to work out what is ##F_Y(y)##?

    ##F_Y(y)## is the probability that ##Y\leq y##. What values of ##X## produce a value of ##Y## that satisfies that? Can you work out the probability that ##X## has such a value?
     
  4. Nov 25, 2015 #3

    mathman

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    By inspection, Y is uniform between 0 and 1.
     
  5. Nov 25, 2015 #4

    WWGD

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  6. Nov 25, 2015 #5

    andrewkirk

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    Unfortunately that only works for increasing or decreasing functions, so it can't be applied to the absolute value function.
     
  7. Nov 25, 2015 #6

    WWGD

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    Why not apply it piecewise, from -1 to 0 then from 0 to 1?
     
  8. Nov 25, 2015 #7
    I'm sorry, can you explain what you just said? I'm tired and feeling slightly brain dead but this homework is due in tomorrow :(

    I've calculated that the pdf is 1/2 for -1<x<1 and 0 otherwise.

    I understand that to find the probability that Y≤y i have to fine the probability that |x|≤y for -1<Y<0 and 0<Y<1 but how do I do that? Am I missing something obvious? Can I just get rid of the modulus because i'm considering all possible values on Y?
     
  9. Nov 25, 2015 #8

    andrewkirk

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    You need to be more careful with upper and lower case, in order to avoid confusing yourself. The standard convention, which works nicely, is to use upper case for a random variable and lower case for a plain old number. So when we write ##Prob(Y\leq y)## we mean the probability that the random variable ##Y## is less than or equal to the number ##y##.
    So don't write things like ##|x|≤y## as you did above, because you'll just confuse yourself. If you instead write ##|X|\leq y## then it's easy to see how to proceed. You are trying to find the probability that the random variable ##X## has an absolute value in the range ##[0,y]##. So ask yourself:
    1. What range does ##X## have to be in for that to be the case?
    2. What is the probability of ##X## being in that range?
     
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