Question

The diameter of a particle of contamination (in micrometers) is modeled with the probability density function for . Determine the following (round all of your answers to 3 decimal places): (a) Enter your answer in accordance to the item a) of the question statement .972 (b) Enter your answer in accordance to the item b) of the question statement .0123 (c) Enter your answer in accordance to the item c) of the question statement .028 (d) Enter your answer in accordance to the item d) of the question statement .972 (e) Determine such that . Enter your answer in accordance to the item e) of the question statement

Answer 1

Complete Question

The complete question is shown on the first uploaded image

Answer:

a
`P(X <  5) =  0.960`

b
`P(X >  8) = 0.016`

c
`P(6 < x < 10) =  0.018`

d
`P(X < 6 or  X > 10 ) =  0.982`

e
`X =  2`

Explanation:

From the question we are told that

The probability density function is
`f(x) =  (2)/(x^3)` for x > 1

Considering question a

`P(x < 5) = \int\limits^5_1 {(2)/(x^3) } \, dx`

=>
`P(X <  5) =  [-(1)/(x^2) ]|  \left \ 5} \atop {1}} \right.`

=>
`P(X <  5) = - (1)/(25)  +   (1)/(1^2)`

=>
`P(X <  5) =  0.960`

Considering question b

`P(x > 8) =1  - \int\limits^6_1 {(2)/(x^3) } \, dx`

=>
`P(X > 8) =1-  [-(1)/(x^2) ]|  \left \ 8} \atop {1}} \right.`

=>
`P(X >  8) = 1 - [- (1)/(64)  +   (1)/(1^2)]`

=>
`P(X >  8) = 0.016`

Considering question c

`P(6 < x < 10) = \int\limits^(10)_(6) {(2)/(x^3) } \, dx`

=>
`P(6 < x < 10) =  [-(1)/(x^2) ]|  \left \ 10} \atop {6}} \right.`

=>
`P(6 < x < 10) =  [- (1)/(100)  +   (1)/(36)]`

=>
`P(6 < x < 10) =  0.018`

Considering question d

`P(X < 6 or  X > 10 ) = 1 - P(6 < x < 10) = 1 - \int\limits^(10)_(6) {(2)/(x^3) } \, dx`

=>
`P(X < 6 or  X > 10 ) =1-  [-(1)/(x^2) ]|  \left \ 10} \atop {6}} \right.`

=>
`P(X < 6 or  X > 10 ) =1- [- (1)/(100)  +   (1)/(36)]` [/tex]

=>
`P(X < 6 or  X > 10 ) =  0.982`

Considering question e

`P(X  <  x ) =  \int\limits^x_1 {(2)/(x^3) } \, dx  =  0.75`

`P(X  <  x ) =  [- (1)/(x^2) ]| \left \ x } \atop {1}} \right.  =  0.75`

`P(X  <  x ) =  - (1)/(x^2) - [- (1)/(1^2) ]= 0.75`

`P(X  <  x ) =  - (1)/(x^2) + 1 = 0.75`

`- (1)/(x^2)  = -0.25`

`X =  2`