If $Y$ is independent of $X_{1}$ and $X_{2}$, does it indicate $Y$ is also independent of $X_{1}+X_{2}$?

We know that $$Y$$ is independent of a vector $$X$$ means it's independent of all linear combinations of components of $$X$$. Then I'm curious about the correctness of the following statement:

If $$Y$$ is independent of $$X_{1}$$ and $$X_{2}$$, $$Y$$ is also independent of $$X_{1}+X_{2}$$.

If this is true, does it mean if we know $$Y$$ is independent marginally of all components of $$X$$, then we'll come to $$Y$$ is independent of $$X$$?

• It depends on whether you read "$Y$ is independent of $X_{1}$ and $X_{2}$" as (a) "$Y$ is independent of $X_{1}$ and $Y$ is independent of $X_{2}$" so pairwise independent rather than as (b) "$Y$ is independent of $X=(X_{1},X_{2})$" which would be jointly independent. With joint independence, $Y$ would also be independent of $X_1+X_2$ Jun 18, 2021 at 15:02
– whuber
Jun 23, 2021 at 17:33
• You should state X1 is independent from X2 else trivial solutions like X1 = -X2; Y = X1 + X2 show up. Aug 1, 2021 at 16:51

Somewhat surprisingly, this is not necessarily true. For example, consider the joint probability distribution described by the following table.

$$Y$$ $$X_1$$ $$X_2$$ prob
0 1 0 0.25
1 1 1 0.25
1 0 0 0.25
0 0 1 0.25

Each of the three random variables follows marginally a Bernoulli($$0.5$$) distribution, and it is very easy to confirm that $$Y \perp X_1$$ and $$Y \perp X_2$$. However, consider that the probability $$P(Y=1, \ X_1 + X_2 = 1) = 0$$ while on the other hand $$P(Y=1)P(X_1+X_2 = 1) = \frac{1}{2}\times\frac{1}{2} = 0.25.$$ Thus, $$Y$$ is not independent of $$X_1 + X_2$$.

• This answer implicitly supposes "independent of $X_1$ and $X_2$" means separately independent of each variable. Many readers will understand this phrase instead to be equivalent to "independent of $(X_1,X_2)$"--but in this latter sense, the conclusion is true, as shown at stats.stackexchange.com/questions/94872. Your example is one in which $Y$ is a function of $(X_1,X_2)$ (with no random error).
– whuber
Jun 23, 2021 at 17:33
• @whuber, I understand your point (which is also captured in a highly upvoted comment on the question). In my opinion, the final sentence of the OPs question makes it clear that this is the intention ("marginally independent"). Jun 23, 2021 at 18:18
• Thank you for pointing out the "marginally" qualifier.
– whuber
Jun 23, 2021 at 18:49