# Solve the following problem : The p.d.f. of the r.v. X is given by f(x) = {12a,for 0< x=2a.0otherwise.Show that P(X<a2)=P(X>3a2) - Mathematics and Statistics

Sum

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).

Is there an error in this question or solution?

#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Probability Distributions
Part I | Q 1.14 | Page 156