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Math Help - Joint Distribution Comprehension Issues

  1. #1
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    Joint Distribution Comprehension Issues

    I am really struggling with comprehending joint distributions. Here is a question that I want to understand. I am not too fussed whether anybody gives me the actual answer!

    Suppose 2 components have independent exponentially distributed lifetimes T_1 and T_2 with parameters \alpha, \beta.
    Find:  \mathbf{P} (T_1 < T_2)

    I understand that the independence of T_1, T_2 allows me to find the joint distribution. So I can say:
     f_{T_1,T_2}(t_1, t_2) = \alpha e^{-\alpha t_1} \beta e^{-\beta t_2}

    I am having a difficult time understanding why to find  \mathbf{P} (T_1 < T_2) I then stick the above in between double integrals and evaluate.
    i.e.,  \mathbf{P} (T_1 < T_2) = \iint \alpha e^{-\alpha t_1} \beta e^{-\beta t_2}dt_1 dt_2

    I am also struggling with the limits on the integrals (not just with their LaTex representation!)
    Can someone english-ify this for me?
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  2. #2
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    Re: Joint Distribution Comprehension Issues

    Anybody?!
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