Probability of a triangle inside a squareIf we have the square with vertices at the 4 corners of ( 0 , 1 ) 2 , and we choose a random point z inside the square, the triangle is between (0,0), (1,0) and z, what is the CDF and PDF of the random variable A T representing the area of the triangle?

Question

Probability of a triangle inside a squareIf we have the square with vertices at the 4 corners of   (  0  ,  1      )    2  , and we choose a random point z inside the square, the triangle is between (0,0), (1,0) and z, what is the CDF and PDF of the random variable       A    T   representing the area of the triangle?

Gaige Burton · Accepted Answer

Step 1Let   (  x  ,  y  )  =  z. The triangle has a base of 1 and a height of y, so thus an area of       A    T    =      1    2    y. Note that       A    T   does not depend on x, only on y.Step 2Your question does not explicitly say what is meant by “a random point”, but I assume that it&#039;s a continuous uniform distribution on the interval (0,1). Since       A    T   is just a constant multiple of a uniformly-distributed variable (y), it itself is uniformly-distributed, but on the interval   (  0  ,      1    2    ).You should be able to work out the CDF and PDF from there.

If we have the square with vertices at the 4 corners of (0,1)^2, and we choose a random point z inside the square, the triangle is between (0,0), (1,0) and z, what is the CDF and PDF of the random variable A_T representing the area of the triangle?

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