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    Distribution Theory

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    Determine the distribution of Y = X_1 + X_2 + . . . + X_n
    by first determining the joint distribution of

    Z_1 = X_1
    Z_2 = X_1 + X_2
    Z_3 = X_1 + X_2 + X_3
    Z_n = X_1 + X_2 + X_3 + . . . + X_n

    and then computing the marginal distribution of Z_n

    According to my text book:

    Exp(a), a>0 has density

    f(x)=(1/a)e^(-x/a), x>0

    and E[X]=a Var(X)=a^2 and the characteristic function is phi x(t)=1/(1-ait)

    This is all the information I have regarding this problem. I hope its enough for someone. I have no idea what to do with this and I need a detailed step by step answer if possible. I really need help on this please!!! I have a test on Tuesday!!

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    Solution Preview

    Here it is given that are independent Exp[a] distributed random variables.
    Now, the probability density function of is given by,

    Thus the joint probability density function of is given ...

    Solution Summary

    The solution describes the determination of the joint distribution of the sum of n independent Exponential random variables.