Let X and Y be the coordinates of a point chosen uniformly at random from the triangle that joins the points (0,1), (0,0), and (1,0).
1) Find the joint distribution of X and Y.
2) Determine the expected value of X and the expected value of Y (these are the expected coordinates of a point chosen at random).
3) Find the correlation between X and Y.
4) If the original units were measured in inches, would there be a different correlation if the units were changed to centimeters? Justify your answer mathematically?
For 1, I got [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]X[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]Y[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT] and 2 I got [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]X[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]y[/FONT] and [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]Y[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT]. Then for correlation in 3 I got [FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]12[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT]. I feel like I am forgetting something however and this should be different. For 4, I know this is unchanged but I am not sure how to show this. Any help is appreciated.
1) Find the joint distribution of X and Y.
2) Determine the expected value of X and the expected value of Y (these are the expected coordinates of a point chosen at random).
3) Find the correlation between X and Y.
4) If the original units were measured in inches, would there be a different correlation if the units were changed to centimeters? Justify your answer mathematically?
For 1, I got [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]X[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]Y[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT] and 2 I got [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]X[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]y[/FONT] and [FONT=MathJax_Math]f[/FONT][FONT=MathJax_Math]Y[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]=[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Math]x[/FONT]. Then for correlation in 3 I got [FONT=MathJax_Main]1[/FONT][FONT=MathJax_Main]12[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]4[/FONT][FONT=MathJax_Math]x[/FONT][FONT=MathJax_Math]y[/FONT][FONT=MathJax_Main])[/FONT]. I feel like I am forgetting something however and this should be different. For 4, I know this is unchanged but I am not sure how to show this. Any help is appreciated.