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Question
Solve the following problem :
The p.d.f. of the r.v. X is given by
f(x) = `{((1)/(2"a")",", "for" 0 < x= 2"a".),(0, "otherwise".):}`
Show that `"P"("X" < "a"/2) = "P"("X" > (3"a")/2)`
Solution
`"P"("X" < "a"/2) = int_0^("a"/2)f(x)*dx`
= `int_0^("a"/2) (1)/(2"a")*dx`
= `(1)/(2"a")[x]_0^("a"/2)`
= `(1)/(2"a")("a"/2 - 0)`
= `(1)/(4)` ...(i)
`"P"("X" > (3"a")/2) = int_((3"a")/2)^(2"a")f(x)*dx`
= `int_((3"a")/2)^(2"a") (1)/(2"a")*dx`
= `(1)/(2"a")[x]_((3"a")/2)^(2"a")`
= `(1)/(2"a")[2"a" - (3"a")/2]`
= `(1)/(2"a") xx "a"/(2)`
= `(1)/(4)` ...(ii)
From (i) and (ii), we get
`"P"("X" < "a"/2) = "P"("X" > (3"a")/2)`.
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