Statistical notation for random variables

Question

I have a question about the notation for the following question given to me by a lecturer.

In this, there are two random variable X and Y, with elements (a,b) and ($\alpha, \beta$) respectively. The question then asks for $p_{XY}(x,y)$. Is this a typo where 'x' should be 'a' and 'y' should be '$\beta$'? I.e. is the question asking for the probability of 'a' and '$\beta$' occurring (0.28), or is it asking for the probabilities of any outcome (see table below).

    |     X       |
    |-------------|
    | a    |   b  |
-|--|-------------|
 |α | 0.12 | 0.18 | 
Y|--|-------------|
 |β | 0.28 | 0.42 |
-|--|-------------|

Answer 1

If $x$ is a typo for $a,$ it would be rather silly to ask you to "calculate" $p_X(x)$ after telling you that $p_X(a) = 0.4,$ wouldn't it?

More likely, $x$ represents either $a$ or $b$ and $y$ represents either $\alpha$ or $\beta,$ so $p_{XY}(x,y)$ represents any of the four probabilities in your table, and you're supposed to calculate all four possibilities (which you have done). Since there are only four of them, it's easy enough to explicitly show each one.

Similarly, $p_X(x)$ and $p_Y(y)$ each have two values that are to be shown.

Answer 2

It is easier to find $p_X(x)$ and $p_Y(y)$ first. We are given that $p_X(a) = 0.4$, therefore $p_X(b) = 1 - p_X(a) = 0.6$. Doing the same thing with $Y$ gives $p_Y(\alpha) = 0.3$.

We also know that $$p_{Y|X}(\alpha|a) = \frac{p_{XY}(a,\alpha)}{p_X(a)} \implies p_{XY}(a,\alpha) = 0.7* 0.4 = 0.28$$ If $X$ and $Y$ are independent, $p_{XY}(x,y) = p_X(x)p_Y(y)$ for any $x,y$. But we have that $$p_{XY}(a,\alpha) = 0.28 \ne p_X(a)p_Y(\alpha) = 0.12$$ Therefore, $X,Y$ are not independent.

As for $p_{XY}(x,y)$, you can find it with the marginals. $$p_{XY}(a,\beta) = p_X(a) - p_{XY}(a,\alpha) = 0.4 - 0.28 = 0.12$$ $$p_{XY}(b,\alpha) = p_Y(\alpha) - p_{XY}(a,\alpha) = 0.3 - 0.28 = 0.02$$ $$p_{XY}(b,\beta) = p_X(b) - p_{XY}(b,\alpha) = 0.6 - 0.02 = 0.58$$