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प्रश्न
If a r.v. X has p.d.f., f(x) = `1/(xlog3)`, for 1 < x < 3, then E(X) and Var(X) are respectively ______
विकल्प
`2/(log3), (4(log3 - 1))/(log3)^2`
`1/(log3), (4(log3 - 1))/(log3)`
`1/(log3)^2, ((log3 - 1))/(4(log3)^2`
`(4(log3 - 1))/(log3)^2, 2/(log3)^2`
उत्तर
If a r.v. X has p.d.f., f(x) = `1/(xlog3)`, for 1 < x < 3, then E(X) and Var(X) are respectively `underline(2/(log3), (4(log3 - 1))/(log3)^2)`
Explanation:
E(X) = `int_-∞^∞ xf(x) = int_1^3 xf(x) dx`
= `int_1^3x 1/(xlog3)dx`
= `1/(log3)int_1^3 1dx = 1/(log3)[x]_1^3`
= `1/log3[3 - 1]`
= `2/log3`
E(X2) = `int_-∞^∞ x^2 f(x)dx`
= `int_1^3 x^2 f(x) dx`
= `int_1^3x^2. 1/(xlog3)dx`
= `1/log3int_1^3 x dx`
= `1/(2log3)[x^2]_1^3`
= `1/(2log3)`[9 - 1]
= `8/(2log3)`
= `4/(log3)`
∴ Var(X) = E(X2) - [E(X)]2
= `4/(log3) - (2/log3)^2`
= `4/((log3)) - 4/(log3)^2`
= `(4log3 - 4)/(log3)^2`
= `(4(log3 - 1))/(log3)^2`
