Is the sum of two variables independent of a third variable, if they are so on their own?

by matthiash   Last Updated January 11, 2019 11:19 AM

Given 3 random variables \$X_1\$, \$X_2\$ and \$Y\$. \$Y\$ and \$X_1\$ are independent. \$Y\$ and \$X_2\$ are independent. Intuitively I would assume that \$Y\$ and \$X_1+X_2\$ are independent. Is this the case, and how can I prove it formally?

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Note that \$X_1 + X_2\$ is a function of \$Z = (X_1,X_2)\$ because if you take \$\$f(x,y) = x+y\$\$ you get \$X_1 + X_2 = f(Z)\$.

It is a well known theorem of probability that if \$R_1\$ and \$R_2\$ are independent random variables and \$f_1\$ and \$f_2\$ are measurable functions then \$f_1(R_1)\$ is independent of \$f_2(R_2)\$ (Theorem 10.4 of "Probability: A Graduate Course" 2nd ed. by Allan Gut).

Since \$f\$ is measurable and Y is independent of \$Z\$ we know that \$Y\$ is also independent of \$f(Z) = X_1 + X_2\$. Note that we took \$f_1\$ as the identity function and \$f_2 = f\$.

It is possible to construct X1, X2, Y such that the above conditions are satisfied, but Y is a function of Z = (X1, X2):

https://math.stackexchange.com/questions/1712177/if-a-random-variable-is-independent-from-the-two-components-of-a-random-vector

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