Let the random variable X is defined as time (in minutes) that elapses between the bell and end of the lecture in case of collagen professor whrer pdf is defined as `f(x)={{:(kx^2","0lexlt2),(0", ""elsewhere"):}`
find the probability that lecture continue for atleast 90s beyond the bell

Options

A. `(37)/(64)`
B. `(35)/(64)`
C. `(33)/(69)`
D. None of these

Answer

A

Explanation

We known that `oversetoounderset(-oo)intf(x)dx=1`
`therefore" "0+underset0overset2intf(x)dx=1`
`rArr" "underset0overset2intkx^2dx=1`
`rArr" "k[x^3/3]_0^2=1rArr[8/3]=1`
`rArr" "k=3/8`
Clearly, the probability that the lecture continuous for at least 90s i.e.`3/2` min beyond the bell
`=P(3/2lexle2)=overset2underset(3//2)intf(x)dx=k""overset2underset(3//2)intx^2dx`
`=k[x^3/3]_(3//2)^2=k/3[8-(27)/8]=(37k)/(24)`
`=(37xx3/8)/(24)=(37)/(64)`