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Find k if the following function represents the p. d. f. of a r. v. X.

f(x) = `{(kx, "for" 0 < x < 2),(0, "otherwise."):}`

Also find `"P"[1/4 < "X" < 1/2]`

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#### Solution

Given that f(x) represents p.d.f of r.v. X.

∴ `int_0^2f(x)*dx` = 1

∴ `int_0^2"k"x*dx` = 1

∴ `"k" int_0^2 x*dx` = 1

∴ `"k"/(2)[x^2]_0^2` = 1

∴ `"k"/(2)[4 - 0]` = 1

∴ `"k"/(2)[4]` = 1

∴ k = `(1)/(2)`

`"P"[1/4 < "X" < 1/2] = int_(1/4)^(1/2)f(x)*dx`

= `int_(1/4)^(1/2) x/(2)*dx`

= `(1)/(2) int_(1/4)^(1/2) x*dx`

= `(1)/(4)[x^2]_(1/4)^(1/2)`

= `(1)/(4)[1/4- 1/16]`

= `(1)/(4)[(4 - 1)/16]`

= `(3)/(64)`.

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