Let p be a random point uniformly distributed in an equilate

Let p be a random point uniformly distributed in an equilateral triangle of area 1 square meter. Let X(p) be the distance from p to the boundary of the triangle. Find the cumulative distribution function Fx of X and the probability density function fx(x). What is the expectation E(X)?

Solution

Area is 1 square feet. Hence prob for p at any point is equal.

If X(p) is the distance from boundary then x(p) can take values from 0 to circumcentre maximum .

For equilateral triangle circum radius = a/2sin60 = a/rt3.

For equil traingle area = rt 3 a^2/4 =1

Hence a = 2/4th root of 3

Circum radius = a/rt 3 = a/ 3^3/4

Thus x varies from 0 to 2/ 3^3/4

Prob x = 1/ b-a where b = 2/ 3^3/4 and a=0

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Cum distribn of X = 0, if x<0

= , if a<=x

=1, for x >=b

where a = 0 and b = 2/ 3^3/4

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E(x) = 1/ 3^3/4