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

i)
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.