Question

SELF ASSESSMENT B A commuter travels into town by KTM train and then has to catch a bus from station to the office. The time, X minutes, that the commuter has to wait for the bus can be modelled by the probability density function: f(x) = 0.5k(6 - x) ,0 SX4 ,4<xs6 otherwise where kis a constant. i. Find the value of k. ii. Sketch the graph of F(x) versus X. iv. Using F(x), show that the probability the commuter has to wait longer than 5.5 minutes for the bus is 0.0125. v. Find the cumulative distribution function of X.

Answer 1

K page tet 0.5K (6-x) otherwise O K s constant First we find vatue of F iven fcx) is probabitity density Funceion ive. I Kdx +fo.s K(6-x) dr =1 jre-) dz i 4 KC4) +0.5K 67- - 1 akto.sk(36-36)- (24-11 4Kto.5K (2) = 1 5K =1 K= 5 have to find graph of F(x) versus x we

page 2 For ony 2,whtre, os4, I K dt dt X For x,where 4s x $6 F(I) dtt o.5x(6-t )dt 2 t0.1[6t-t 2 .8.(-2) 0.8 +0.1 6Y 0.3+0.1(61- -1 2 - o.051-1.G 0.8 +0.6 --0.8+06T - 0.05x FCx 0.8+0.6 x- 0.05 otherwise of Fox) versus X The is graph F(x 1-0 O-8 0.6 0 2

Poge 3 we have to show, P(x>5.s) = o.0125 Ty Consides -P(x < 5.5) Pix5.5) 1-F(5.5) =i-8t0.6 (5.5) - 0.05 (5-5) = 1-0.987S = 0.0125 :P (x> 5.5)- 0.D125 Hence proved. Cumutative distribution fun is, FOX) x)dx For any where. 0sx4 X FOx) dt For any x, where F(x) dtSo1 (6-t)dt X 0.8 + 0.1 6t- =a8+0.1(61- (24-8)) 2 0.8to-6X-0.05I 1e. F(X) -0-8+0.6.X -O.050, 4 Sx56