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I am interested to know how to calculate the joint probability mass function for two independent geometric random variables.

Suppose two variables X1 and X2 are independent, such that Xi∼Geometric(theta), how to find the joint pmf distribution of X1 and X2. I am not sure but I think it should be the product of pmf of both mass function.

Also, how should I calculate the probability of the event where kth trial being the the first success/failure for both the variables or k1th trial for X1 and k2th trial for X2?

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    $\begingroup$ As you mentioned, the joint pdf of two random variables is $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ if they are independent. So if $X,Y\overset{iid}{\sim}\text{Geometric}(p)$ then $f_{X,Y}(x,y) = (1-p)^{x-1}p(1-p)^{y-1}p = (1-p)^{x+y-2}p^2$ $\endgroup$
    – bdeonovic
    Oct 27, 2013 at 4:09

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Answered in comments:

As you mentioned, the joint pdf of two random variables is $f_{X,Y}(x,y)=f_X(x)f_Y(y)$ if they are independent. So if $X,Y\overset{iid}{\sim}\text{Geometric}(p)$ then $f_{X,Y}(x,y) = (1-p)^{x-1}p(1-p)^{y-1}p = (1-p)^{x+y-2}p^2$

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