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प्रश्न
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]`
उत्तर
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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