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Question
If a r.v. X has p.d.f f(x) = `{("c"/x"," 1 < x < 3"," "c" > 0),(0"," "otherwise"):}`
Find c, E(X), and Var(X). Also Find F(x).
Solution
a. Given that f(x) represents p.d.f. of r.v. X
∴ `int_1^3 f(x)*"d"x` = 1
∴ `int_1^3 "c"/x*"d"x` = 1
∴ `"c" int_1^3 (1)/x*"d"x` = 1
∴ `"c"[logx]_1^3` = 1
∴ c [log 3 – log 1] = 1
∴ c [log 3 – 0] = 1
∴ c = `(1)/log3`
b. E(X) = `int_(-oo)^(oo) xf(x)`
= `int_1^3 xf(x)*"d"x`
= `int_1^3 x "c"/x*"d"x`
= `"c" int_1^3 1*"d"x`
= `(1)/log3 [x]_1^3`
= `(1)/log3[3 - 1]`
= `(2)/log3`.
c. E(X2) = `int_(-oo)^(oo) x^2f(x)`
= `int_1^3 x^2f(x)*"d"x`
= `-int_1^3 x^2. "c"/x*"d"x`
= `"c" int_1^3x*"d"x`
= `(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`
F(x) = `int_1^x f(x)*"d"x`
= `int_1^x "c"/x*"d"x`
= `"c" int_1^x (1)/x*"d"x`
= `"c"[logx]_1^x`
= c[log x – log 1]
= c log x
= `log x/log 3`
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