Let (X,Y) have uniform distribution on the four points(0,1),(0,−1),(1,0),(−1,0). How can I show that X and Y are uncorrelated but not independent? Could someone just point me in the right direction for this problem. I don't know how to start here
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2 Answers
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This is a clever study in contingency tables. Build the contingency table.
Each variable has support $\{-1,0,1,\}$, so you need to draw a $3 \times 3$ table. Put inside the probabilities you are given, and fill the other cells too, so that the whole satisfies the laws of probability. Remember, or read about, how from the contingency table that holds the joint probabilities we can get the marginal probabilities (hint: it is from here that they acquired the adjective "marginal"). Then apply the advise given to you by the other answer, to show what you are asked to show.
answered May 28, 2020 at 23:17
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For uncorrelatedness, you just calculate $E[XY]-E[X]E[Y]$, which can be calculated via simple double sum. For example, the first term can be calculated as:
$$E[XY]=\sum_x\sum_y xy p_{XY}(x,y)\rightarrow 0$$
For independence, try to find $x_0,y_0$ such that $p_{XY}(x_0,y_0)\neq p_{X}(x_0)p_Y(y_0)$. For example, $y_0$ can be $0$.
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$\begingroup$ So E[X] = Sum x to infinity of xP_x(x)-> 0 ? $\endgroup$– No NimeCommented May 29, 2020 at 18:52
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$\begingroup$ Yes, expectations are $0$ as well. $\endgroup$– gunesCommented May 29, 2020 at 19:05
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$\begingroup$ So, are we saying that E[XY]−E[X]E[Y] just = 0? does that just imply that its uncorrelated in itself? Also, I'm confused about the independence statement... how do set up something to find that? $\endgroup$– No NimeCommented May 29, 2020 at 20:28
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$\begingroup$ It's the definition of covariance, if covariance is $0$ RVs are uncorrelated. For independence, put some values for $x_0,y_0$ (i.e. -1,0,1) and see if the inequality holds for a case. $\endgroup$– gunesCommented May 29, 2020 at 21:20
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$\begingroup$ i understand that, but how do i just solve for pXY(x0,y0) or pXY(x0,y0)≠pX(x0)pY(y0) by plugging in? $\endgroup$– No NimeCommented May 29, 2020 at 23:39
self-studytag if this is some sort of homework. $\endgroup$