Each rear tire on an experimental airplane issupposed to be filled with a pressure of 40 pound per squareinch (psi). Let X denote the actual air pressure for the right tireand Y denote the actual air pressure for the left tire. Suppose that Xand Y are random varibles with the jointdensity

f (x, y) = k (x^2+y^2, 30<-x<50;

30<-y<50

0, elsewhere

a.) find k

b.) find P (30<-x<-40 and 40<-Y<50)

c.) Find the probability that both tires areunderfilled.

show all steps to solve ... thx

a. K = 3/3920000

b. 0.26

c. 0.189

Step-by-step explanation:

f (x, y) is a join distribution, so the following condition must be satisfied

∫∫ f (x, y) dydx {-∞ to ∞} {-∞ to ∞} = 1

A. Finding K

∫∫ k (x² + y²) {30 to 50}{30 to 50} dydx = 1

k ∫∫ (x² + y²) {30 to 50}{30 to 50} dydx = 1

k ∫ (x²y + y³/3) {30 to 50}{30 to 50} dx = 1

k ∫ (50x² + 50³/3 - 30x² - 30³/3) dx {30 to 50} = 1

k ∫ (20x² + (50³ - 30³) / 3) dx {30 to 50} = 1

k ∫ (20x² + 98000/3) dx {30 to 50} = 1

k (20x³/3 + 98000x/3) {30,50} = 1

k (20 * 50³/3 + 98000 * 50/3 - 20 * 30³/3 - 98000 * 30/3) = @

k (3920000) / 3 = 1

K = 3/3920000

b.

P (30<-x<-40 and 40<-Y<50) = ∫∫ k (x² + y²) {30 to 40}{40 to 50} dydx

k ∫∫ (x² + y²) {30 to 40}{40 to 50} dydx

k ∫ (x²y + y³/3) {30 to 40}{40 to 50} dx=

k ∫ (50x² + 50³/3 - 40x² - 40³/3) dx {30 to 40}

k ∫ (10x² + (50³ - 40³) / 3) dx {30 to 40}

k ∫ (10x² + 61000/3) dx {30 to 40}

k (10x³/3 + 61000x/3) {30,40}

k (10 * 40³/3 + 61000 * 40/3 - 10 * 30³/3 - 61000 * 30/3)

k (980000/3)

But k = 3/3920000

So

P (30<-x<-40 and 40<-Y<50) = 3/3920000 * 980000/3

P (30<-x<-40 and 40<-Y<50) = 0.25

c. Probability that both tries are undefined.

This means that both pressure are between 30 and 40.

So, we'll solve for P (30<-x<-40 and 30<-Y<40)

P (30<-x<-40 and 30<-Y<40) = ∫∫ k (x² + y²) {30 to 40}{30 to 40} dydx

k ∫∫ (x² + y²) {30 to 40}{30 to 40} dydx

k ∫ (x²y + y³/3) {30 to 40}{30 to 40} dx=

k ∫ (40x² + 40³/3 - 30x² - 30³/3) dx {30 to 40}

k ∫ (10x² + (40³ - 30³) / 3) dx {30 to 40}

k ∫ (10x² + 37000/3) dx {30 to 40}

k (10x³/3 + 37000x/3) {30,40}

k (10 * 40³/3 + 37000 * 40/3 - 10 * 30³/3 - 37000 * 30/3)

k (740000/3)

But k = 3/3920000

So

P (30<-x<-40 and 30<-Y<40) = 3/3920000 * 740000/3

P (30<-x<-40 and 30<-Y<40) = 0.188775510204081

P (30<-x<-40 and 30<-Y<40) = 0.189