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Math Help - Find probability density function

  1. #1
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    Find probability density function

    Suppose that X and Y are independent and each is uniformly distributed on (0, 1). Let U = X +
    Y and V = X - Y.
     Sketch the range (the region of non zero probability) of (X, Y) and the range of (U, V).
     Find the density function of (U, V).
     Find the density function of U.
     Find the density function of
    I'm not really sure where to start with this question. By being uniform, I assume it means it's mean is 0 and SD is 1? And a normal distribution?
    But I couldn't find any information in terms of adding or subtracting them, partly I guess because I have a lot of math to revise for this to make more sense.

    If there are any suggestions, or if anyone could prod me in the right direction, i'd appreciate it! =)
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  2. #2
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    Re: Find probability density function

    Hey AshleyCS.

    A uniform distribution without any extra information is basically a PDF of P(X = x) = 1 for 0 < x < 1 and 0 everywhere else.

    You can find the individual density functions using convolution theorems in probability. With regards to the joint distributions, you should consider all pairs of values for x and y for functions u and v.

    A few identities that will be useful will include P(U = u, V = v) = P(U=u|V=v)*P(V=v) = P(V=v|U=u)*P(U=u). You can get the conditionals by comparing the different distributions you obtained above.

    How much probability have you covered? What topics are you currently studying?
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  3. #3
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    Re: Find probability density function

    Quote Originally Posted by chiro View Post
    Hey AshleyCS.

    A uniform distribution without any extra information is basically a PDF of P(X = x) = 1 for 0 < x < 1 and 0 everywhere else.

    You can find the individual density functions using convolution theorems in probability. With regards to the joint distributions, you should consider all pairs of values for x and y for functions u and v.

    A few identities that will be useful will include P(U = u, V = v) = P(U=u|V=v)*P(V=v) = P(V=v|U=u)*P(U=u). You can get the conditionals by comparing the different distributions you obtained above.

    How much probability have you covered? What topics are you currently studying?
    Hey thanks for the reply!

    Have covered quite a lot in previous years, i'm doing a CS course so just havn't touched on much maths in the last 2 years and have then taken a very mathmatical course so lots of revision!

    Thankyou for your response, it gives me a good start for looking at what other math I need to go through!
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