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.

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