Please help me solve this question and explain into detials. I am told we will have to draw a triangle but I dont know how. Can you help with this please?     A point v = (x, y) is chosen uniformly at random from the triangular region with vertices at the points (0, 0), (2, 0), and (0, 2). Let A denote the event that x ≤ y. Let B denote the event that x ≤ 1. Let C denote the event that x + y ≤ 1. (a) Are A and B independent? Why, or why not? (b) Are A and C independent? Why, or why not?

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Please help me solve this question and explain into detials. I am told we will have to draw a triangle but I dont know how. Can you help with this please?

 

 

A point v = (x, y) is chosen uniformly at random from the triangular region with vertices at the points (0, 0), (2, 0),
and (0, 2).
Let A denote the event that x ≤ y.
Let B denote the event that x ≤ 1.
Let C denote the event that x + y ≤ 1.
(a) Are A and B independent? Why, or why not? (b) Are A and C independent? Why, or why not?

Expert Solution
Step 1: Independence of Events shown.

The triangle is ABC whose area is 1 half left parenthesis A B x A C right parenthesis equals 4 over 2 equals 2

P(Xless or equal than Y) = fraction numerator A r e a space A C D over denominator A r e a space A B C end fraction equals 1 half= 0.50 =P(A)

Probability homework question answer, step 1, image 1

P(Xless or equal than1) =fraction numerator A r e a space A E D C over denominator A r e a space A B C end fraction equals fraction numerator A r e a space A E D F space plus space A r e a space F D C over denominator A r e a space A B C end fraction equals fraction numerator 1 plus 0.5 over denominator 2 end fraction equals fraction numerator 1.5 over denominator 2 end fraction

=0.75 = P(B)

Probability homework question answer, step 1, image 2


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