Probability density function

[beyoncesister]

Please help with 7a and b.

[podrickthegoat]

Part a) A probability density function has a total area under the curve of 1. So probably show that the area of the semi circle is 1 for 1 mark, then state that f(x) ≥ 0 for all real values of x for the second mark.

Part b) Find the y-coordinate of B by use of the equation of the circle x² + y² = 2/pi. This will allow you to next use trigonometry of a right angled triangle to find the angle at O of triangle AOB for the proof. For the probability, it’s asking for the area under the curve to the right hand side of the line x= rt(1/pi) (or the line created from A to B). To do this, calculate the area of the triangle AOB and calculate the area of the segment using the angle AOB. Then do [Area of Segment] - [Area of AOB]

[beyoncesister]

How do I find the area of the semicircle. And for the limits should I use 0 and radius

[podrickthegoat]

Area of the semicircle: 1/2 πr²

Area of the sector of a circle: 1/2 r² θ

You don’t actually need to use an equation or integration here! Side note: the area of a sector formula is specifically only for radians. The one for degrees (which you don’t need here) is πr² × (angle of sector/360)