Question

An environment engineer measures the amount ( by weight) of particulate pollution in air samples ( of a certain volume ) collected over the smokestack of a coal-operated power plant. Let X1 denote the amount of pollutant per sample when a certain cleaning device on the stack is not operating, and let X2 denote the amount of pollutant per sample when the cleaning device is operating under similar environmental conditions. It is observed that X1 is always greater than 2X2, and the relative frequency behavior of (X1, X2) can be modeled by

f(x,y)= k for 0 <= x <= 2, 0<=y <=1 , 2y<= x and 0 elsewhere

(X and Y are randomly distriibutied over the region inside the tricanle bounded by x=2, y=0 and 2y=x)

a. Find the value of k that makes this a probability desnsity function.
b. Find P >= 3y

Answer 1

`k = 1`

`P(x > 3y) = (2)/(3)`

Explanation:

Given

`f \left(x,y \right) = \left{ \begin{array} { l l } { k , } & { 0 \leq x} \leq 2,0 \leq y \leq 1,2 y \leq x } & { \text 0, { elsewhere. } } \end{array} \right.`

Solving (a):

Find k

To solve for k, we use the definition of joint probability function:

`\int\limits^a_b \int\limits^a_b {f(x,y)} \, = 1`

Where

`{ 0 \leq x} \leq 2,0 \leq y \leq 1,2 y \leq x }`

Substitute values for the interval of x and y respectively

So, we have:

`\int\limits^2_(0) \int\limits^(x/2)_(0) {k\ dy\ dx} \, = 1`

Isolate k

`k \int\limits^2_(0) \int\limits^(x/2)_(0) {dy\ dx} \, = 1`

Integrate y, leave x:

`k \int\limits^2_(0) y {dx} \, [0,x/2]= 1`

Substitute 0 and x/2 for y

`k \int\limits^2_(0) (x/2 - 0) {dx} \,= 1`

`k \int\limits^2_(0) (x)/(2) {dx} \,= 1`

Integrate x

`k * (x^2)/(2*2) [0,2]= 1`

`k * (x^2)/(4) [0,2]= 1`

Substitute 0 and 2 for x

`k *[ (2^2)/(4) - (0^2)/(4) ]= 1`

`k *[ (4)/(4) - (0)/(4) ]= 1`

`k *[ 1-0 ]= 1`

`k *[ 1]= 1`

`k = 1`

Solving (b):
`P(x > 3y)`

We have:

`f(x,y) = k`

Where
`k = 1`

`f(x,y) = 1`

To find
`P(x > 3y)`, we use:

`\int\limits^a_b \int\limits^a_b {f(x,y)}`

So, we have:

`P(x > 3y) = \int\limits^2_0 \int\limits^(y/3)_0 {f(x,y)} dxdy`

`P(x > 3y) = \int\limits^2_0 \int\limits^(y/3)_0 {1} dxdy`

`P(x > 3y) = \int\limits^2_0 \int\limits^(y/3)_0 dxdy`

Integrate x leave y

`P(x > 3y) = \int\limits^2_0 x [0,y/3]dy`

Substitute 0 and y/3 for x

`P(x > 3y) = \int\limits^2_0 [y/3 - 0]dy`

`P(x > 3y) = \int\limits^2_0 y/3\ dy`

Integrate

`P(x > 3y) = (y^2)/(2*3) [0,2]`

`P(x > 3y) = (y^2)/(6) [0,2]\\`

Substitute 0 and 2 for y

`P(x > 3y) = (2^2)/(6) -(0^2)/(6)`

`P(x > 3y) = (4)/(6) -(0)/(6)`

`P(x > 3y) = (4)/(6)`

`P(x > 3y) = (2)/(3)`