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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"):}`
Find the value of k.

Text Solution

Verified by Experts

Since `f(x)` is p.d.f
`underset(-oo)overset(oo)intf(x) dx = 1`
`thereforeunderset(-oo)overset(0)intf(x)dx + underset(0)overset(2)intf(x)dx +underset(2)overset(oo) f(x)dx = 1`
`therefore 0+underset(0)overset(2)intf(x) dx + 0 = 1 " "....[becausef(x) = 0,"when" x lt 0 and x gt 2]`
`thereforeunderset(0)overset(2)int kx^(2) dx = 1`
` therefore kunderset(0)overset(2)int x^(2) dx = 1`
`therefore k [(x^(3))/(3)]_(0)^(2) = 1" "therefore(k)/(3)[x^(3)]_(0)^(2) = 1`
`therefore(k)/(3)(8-0)=1 " "therefore(8)/(3)k= 1`
`therefore k = (3)/(8)`
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