Independence of random variables where probability = 1

Question

If we have random variables $X,Y$ and we know that $P(X=5)=1$, can we immediately show that $X, Y$ are independent? Intuitively it seems that this information is enough for us to say that we don't need $Y$ to know $X$, but couldn't $Y$ be a function of $X$? Then the joint PMF $P(X,Y) = P(X)P(Y)$, but also $= P(Y)$. How can we justify independence in such cases? Mathematically it works, but it seems we're claiming that a random variable is independent to itself.

Answer 1

Constant or almost surely constant random variables are independent of any other random variable (including itself!) This follows from the fact that an event that has probability 0 or 1 is independent of any other event (including itself!). Let us see why:

Independence of events $A$ and $B$ is defined as $P(A \cap B) = P(A)P(B)$

Now, independence of random variables $X$ and $Y$ can be defined in many ways (in elementary probability theory). One way is that the joint pdf splits up (assuming $X$ and $Y$ have pdfs and a joint pdf) and another is that the joint cdf splits up:

$$P(X \le a)P(Y \le b) = P(X \le a, Y \le b) \tag{*}$$

for $a,b \in \mathbb R$

Define

$$A = \{X \le a\}$$ $$B = \{Y \le b\}$$

Then we have a familiar form:

$$P(A)P(B)=P(A \cap B) \tag{**}$$

I believe you can handle the rest from the second link above but just in case...

We are given that $X$ is constant. Thus for any $a \in \mathbb R$, $P(A)=1$ or $P(A)=0$. If $P(A)=0$, then $P(A \cap B)=0$. So both sides of $(**)$ are 0. If $P(A)=1$, then $P(A^C)=0 \to P(A^C \cap B) = 0 \to P(A \cap B) = P(B) \to (**)$ Hence, $A$ and $B$ are independent for any $a,b \in \mathbb R$

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As for your other concern about $Y$ being a function of $X$:

If $X$ is (almost surely) constant then so is $Y$.

For example if $P(X=5)=1$ and $Y=5X^2+\sin(X)$ then $P(Y = 5X^2+\sin(X) = 125+\sin(5)) = 1$.

Answer 2

If $P(X=x)=1$ then $P(X \leq a,Y \leq b)=\begin{cases} P(Y \leq b) & a \geq x \\ 0 & a<x \end{cases}$ (why?), which is equal to $P(X \leq a)P(Y \leq b)$ regardless of what $a,b$ are (again, why?). It follows that a constant random variable is indeed independent of itself.