Question

Consider the triangle with vertices (0,0),(3,0),(0,3) and let P be a point chosen uniformly at random inside the triangle. Let X be the distance from P to (0,0). (i) Is X a random variable? Explain why or why not. (ii) Compute P(X≥0), P(X≥1), P(X≥2), and P(X≤3).

Answer #1

Yes X is a random variable, and a **continuous** one
because it can take any value beginning from 0.

ii)

X is the distance of point P from (0,0)

P(X ≥ 0) = **1**, since X denotes distance which will
always be positive or zero.

P(X ≥ 1):

In the figure attached, we have sketched an arc representing the
points which are at a distance 1 from (0,0).

Since point P is chosen uniformly inside the triangle,

Area inside arc =
units

Area of triangle = 0.5(3)(3) = 4.5 units

P(X < 1) = 0.1745

P(X≥1) = 1 - 0.1745 = **0.8255**

Similarly P(X≥2) = 1 - P(X < 2)

P(X < 2) = 0.698

P(X≥2) = 1 - 0.698 = **0.302**

P(X≤3) = **1** , since anywhere inside the traingle
the distance of point P from (0,0) is less than 3.

5. Pick a uniformly chosen random point inside the triangle with
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(b) (b) What is the probability that the distance of this point
to the origin is more than 1?

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