Question

5. Let X be a random variable with PDF 30 20 f(x)- 20 < x < 40 0 otherwise. (a) Find P(X 20) and P(X >20) (b) Suppose that buses go past my stop at exactly twenty minutes past the hour and forty every hour. I arrive at my stop at a completely random time during the day. What is the expected value of the length of time I'll have to wait for a bus?

Answer 1

5a)P(X<=20)=$\int_{0}^{20}$ f(x) dx =$\int_{0}^{20}$ (1/30 ) dx =(x/30) |²⁰₀ =20/30=2/3

P(X>20)=1-P(X<=20)=1-2/3 =1/3

b)

here let x is the time when person arrive the stop

therefore f(x) =1/40   if 0 <x<20 or 40<x <60

also f(x)=1/20    for 20<x<40

hence E(X)=(0+40)/2+(0+20)/2=30 minutes