# Distribution Theory

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.