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)`

Answer 1

`"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)`.