The diameter of a particle of contamination (in micrometers) is modeled with the probability density function f(x)= 2/x^3 for x > 1. Determine the following (round all of your answers to 3 decimal places):

a) P(x < 5)=

b) P(x > 8)=

c) P(6 < x < 10)=

d) P(x < 6 or x > 10)=

3) Determine x such that P(X < x) = 0.75

a) 0.96

b) 0.016

c) 0.018

d) 0.982

e) x = 2

Step-by-step explanation:

We are given with the Probability density function f(x)= 2/x^3 where x > 1.

Firstly we will calculate the general probability that of P(a < X < b)

       P(a < X < b) =   =

                            =    { Because }

                            = =

                            = =

a) Now P(X < 5) = P(1 < X < 5)  {because x > 1 }

     Comparing with general probability we get,

     P(1 < X < 5) = = = 0.96 .

b) P(X > 8) = P(8 < X < ∞) = 1/ - 1/∞ = 1/64 - 0 = 0.016

c) P(6 < X < 10) = = = 0.018 .

d) P(x < 6 or X > 10) = P(1 < X < 6) + P(10 < X < ∞)

                                = + (1/ - 1/∞) = 1 - 1/36 + 1/100 + 0 = 0.982

e) We have to find x such that P(X < x) = 0.75 ;

               ⇒  P(1 < X < x) = 0.75

               ⇒   = 0.75

               ⇒   = 1 - 0.75 = 0.25

               ⇒   =   ⇒ = 4 ⇒ x =  

Therefore, value of x such that P(X < x) = 0.75 is 2.