4. (30pts) Choose a point uniformly at random to be one of the four vertices of...

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4. (30pts) Choose a point uniformly at random to be one of the four vertices of the diamond below. Let (X,Y) denote its coordinates, where X and Y are random variables that denote the horizontal and vertical coordinate respectively. (a) Find the PMF of X by specifying the possible values and the corresponding probabilities. (b) Find E(X), E(X+Y) and E(XY). (c) Are X and Y independent? Justify your answer mathematically. (d) Now let's rotate the diamond by 45°. Consider the following square and choose a point uniformly at random to be one of its four vertices. Let (U, V) denote its coordinates. Are U and V independent in this case? Justify your answer mathematically. 4. (30pts) Choose a point uniformly at random to be one of the four vertices of the diamond below. Let (X,Y) denote its coordinates, where X and Y are random variables that denote the horizontal and vertical coordinate respectively. (a) Find the PMF of X by specifying the possible values and the corresponding probabilities. (b) Find E(X), E(X+Y) and E(XY). (c) Are X and Y independent? Justify your answer mathematically. (d) Now let's rotate the diamond by 45°. Consider the following square and choose a point uniformly at random to be one of its four vertices. Let (U, V) denote its coordinates. Are U and V independent in this case? Justify your answer mathematically.

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