Calculus...?

Question

The probability density function of X, the lifetime of a certain type of device (measured in months), is given by

f(x) = 0 if x < 22
f(x) = 22 / x^2 if x > 22

Find the following:
P(X > 45) = ______

The cumulative distribution function of X:
F(x) = Answer 1 if x < 22
F(x) = Answer 2 if x > 22

Answer 1

F(x) = ∫f(x)dx

For 0 ≤ x ≤ 22
F(x) = f(x) = 0
______________

For x ≥ 22, integrate from 22 to x.

F(x) = ∫f(x)dx = ∫(22/x²)dx = -22/x | [Evaluated from 22 to x]

F(x) = (-22/x + 22/22) = 1 - 22/x

P(x > 45) = 1 - F(45) = 1 - (1 - 22/45) = 22/45

Answer 2

P(X > 45) = integral {limits 45 to infinity } f(x)dx

= integral {limits 45 to infinity } [22 / x^2 ]dx

= {limits 45 to infinity } [-22 / x ]

= 0 - [-22 / 45 ]

=22/ 45