Let X be a continuous random variable with the probability density function(p.d.f.) given by f(x) = cases kx, & 0 x P(2 < X 3) =

Step-by-step solution

Since f(x) is a probability density function, the total probability must be 1 . _ - ^ f(x) dx = 1 ₀¹ kx dx + ₁² k dx + ₂³ (-kx + 3k) dx = 1 k [ x^2 2 ]₀^1 + k [x]₁^2 + k [ - x^2 2 + 3x ]₂^3 = 1 k 2 + k(2 - 1) + k ( (- 9 2 + 9 ) - (- 4 2 + 6 ) ) = 1 k 2 + k + k ( 9 2 - 4 ) = 1 k 2 + k + k 2 = 1 2k = 1 k = 1 2 Now, we need to find P(2 P(2 P(2 Substituting k = 1 2 , we get: P(2 Answer: 1 4