Probability Density Function Challenge

[bloodhound]

This was one of my questions from my assingment. pretty nifty.. also wanna see if my answer is consistent with yours.

 

 

In a circle radius a and centre O, a chord PQ is drawn. Suppose that the direction of the chord is chosen at random, that is, the angle [math]\theta[/math] between PQ and the tangent to the circle at P is chosen at random with a uniform distribution over the interval[math][0,\frac{\pi}{2}][/math]. Let the random variable Z denote the area of the traingle POQ. Derive the probability density function of Z

 

 

good luck

[julien]

I obtained [math]P(Z\in[z,z+dz])=\frac{4}{\pi a^2\sqrt{1-\frac{4z^2}{a^4}}}dz[/math] with [math] z\in [0;\frac{a^2}{2}][/math]

 

what about your solution ?

[bloodhound]

yeap, same answer.

 

[math]f_{Z}(z)=\frac{4}{\pi a^2\sqrt{1-\frac{4z^2}{a^4}}}[/math]

 

does it still work if u cancel our the a^2 bits from top and bottom?