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Let X = time (in minutes ) that lapses...

Let X = time (in minutes ) that lapses between the bell and the end of the lectures in cases of a collge professor. Suppose X has p.d.f
`f(x) ={{:(kx^(2),0 le x le 2),(0,"otherwise"):}`
What is the probability that lecture ends within 1 minute of the bell ringing ?

Text Solution

Verified by Experts

Required probability `=P(X le 1) = underset(-oo)overset(1)intf(x)dx`
`=underset(-oo)overset(0) f(x)dx + underset(1)overset(0)intf(x)dx`
`= 0 +underset(0)overset(1)intf(x)dx" "....[because f(x) = 0 , "where" x lt 0]`
`=underset(0)overset(1)intkx^(2)dx = k underset(0)overset(1)intx^(2) dx `
`= k[(x^(3))/(3)]_(0)^(1) =(k)/(3)[x^(3)]_(0)`
`=(k)/(3)[1-0]=(k)/(3)=(1)/(3)xx(3)/(8) =(1)/(8)`
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