Question

A point is selected randomly inside a circle with a radius of R. X denotes the distance between the selected position to circle's center. First, calculate the probability function of X, then find the best results below for the P( R/3 < X < R/2). (Hint: First you should calculate the distribution function of X).

Select one:

a. 4/19

b. 12/41

c. 4/23

d. 5/36

Answer 1

The point is Uniformly distributed the centre of the circle denote the origin and define X and Y to be solution withine LET the circle. point chosen fentent C { -- the co-ordinates of the soint density function of Z and y 22+ y²LR z²+y²7R for some value of we know sosa franys dedy = 1 as Sy² ty? R c dzdy = 1 = CSS dady=1 2²+ y²=R ss dzdy is the area 2² ty ZR of the circle with Radius R which is IR?} thus CAR² = 9 = C = 1 R2 thus feziy) ={ the 9 z²ty?ck 9 z²ty?>R marginal of x feldo= roof anys dy = f TR2 dy freer? TR2 JR²_x² fq (98) = 1 12 [VR ² 22 - VR 2 -JR²2 -2² for 2²LR? feth) 2 FR²-22 for ² <R2 *R? CScanned with CamScanner

Now, the Distribution for D= [q²ty2 the distance from the origin is obtained as follows for OSXER (xi- distance between the selected position to Circle centre) Fore= p ( Voyez +42470 = P(x²+4² 24²) Fylgo = Shp?ty2= 78 Frez dedy Folg deady = area of code) F G = 4² I TR2 (182) (Saeth eg R2 New pdf f= F (q) = Focal da fixe = 2x 9 OLXER # R2 Pdf as X R/2 22 dx Now R2 P( R/ CXC8/2) = R/3 P(R/3 CX CR/2) = b PCRI3CXCR/2) P(R/ CX CR/2) = { Řz [x²] Rl R/3 [ { / - R2 R2 36 option :- (d)=5 36 CScanned with CamScanner