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Can someone help me get started in finding the PDF/CDF of a random variable $$Y = \min({X_1 + X_4, X_1 + X_3, X_2 + X_3, X_2 + X_4}),$$ where $$ X_1, X_2, X_3, X_4 $$ are independent, not necessarily I.I.D though.

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Can someone help me get started in finding the PDF/CDF of a random variable

Well, to start, $\min\{X_1+X_3, X_1+X_4, X_2+X_3, X_2+X_4\}=\min\{X_1,X_2\}+\min\{X_3,X_4\}$

Now $f_{\small\min\{U,V\}}(t)$ equals what, when $U,V$ are independent, and $f_{\small P+Q}(r)$ equals what, when $P,Q$ are independent?

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