Question

Q: A car dealer sells X cars each day and always tries to sell an extended warranty on each of these cars. Let Y be the number of extended warranties sold, then Y >= X. The joint pmf of X and Y is given by:

**Please give the step by steps with details to completely see how the solution came about.

1) The joint pmf of X and Y is given by: f(x,y) = c(x+1)(4-x)(y+1)(3-y), x=0,1,2,3 y=0,1,2 with y $\leq$ x.

    (a) Find the value of c. (Answer is c=1/154 from back of book NOT 1/200) details is what I would like to see, thanks.

    (b) Sketch the support of X and Y.

    (c) Record the marginal pmfs f_x(x) and f_y(y) in the 'margins.

    (d) Are X and Y independent?

    (e) Compute $\mu$ _x and $\sigma$ ²x.

    (f) Compute $\mu$ _y and $\sigma$ ²y.

    (g) Compute Cov(X,Y).

    (h) Determine $\rho$ , the correlation coefficient.

    (i) Find the best-fitting line and draw it on your figure.

Answer 1

ANSWER:

A car dealer sells X cars each day and always tries to sell an extended warranty on each of these cars. (In our opinion, most of these warranties are not good deals.) Let Y be the number of extended warranties sold; then Y ≤ X. The joint pmf of X and Y is given by

х,у x 0,1,2,3, y-0,1,2, with y sx

(a) Find the value of c.

(b) Sketch the support of X and Y.

(c) Record the marginal pmfs f_X(x) and f_Y(y) in the “margins.”

(d) Are X and Y independent?

(e) Compute μ_X and σ²_X.

(f) Compute μ_Y and σ²_Y

(g) Compute Cov(X,Y).

(h) Determine ρ, the correlation coefficient.

(i) Find the best-fitting line and draw it on your figure.

(a) Recall that sinceis a p.m.f., it must be true that

f(x,y)

That is

We will use that equation to find the value of c.

1. Rewrite the previous equation

   
   (x, y) =
   ,
   x=0.1.2.3, y=0.1.2
   

2. Recall that

   

   x(x 0+1+2+ + x =

   

   02 +12 +24 +12 =

3. Using the equations from the previous parts, we get

   

   x=0 y:0 x=0 y=0 x=0 = Σc(x + 1)(4-х).th-(-2x2 +5x + 18) (x+D2 -12c + 42c + 60c + 40c = 1 54c

   Since, we get
   154
   .

   (b) The support of X and Y is shown below.

4. 

5. (c) The marginal p.m.f. is given by following equations

   KCr) ภ์(x)

   

   y-0 , 2 (x+1)(4-x)Σ(y+1)(3-y), x=0, 154 ー (x + 154-x) +143-y), x=3 154 y-

   Using the equation from the previous part, we get

   

   f(0) =-4 154 154

6. 
   -AL 65 65 27 4-7
   

   

   154 v-O 13+4+3) 154 30

   

   4 154y y-0 4 154 20 (3 + 4 + 3) -

7. The marginal p.m.f. is given by the following equations

   /2(v)

   

   ˊ丿, x-y 154 154

   Using the equation from the previous part, we get

   

   154 (4+6+6+4) 154 30

   

   4 154 4 154 32 (6+6+4)

   

   154+4x) =154(6 + 4) r=2 15

   (d) Recall that X and Y are independent if for all values of random variables X and Y it is true that

8. .

   f( X, Y)=f(X)f, (Y)

   Since

   

   fa. 12 154 3042 o,(2) 154 154

   Therefore X and Y are dependant.

   (e) The mean of X is given by

9. 

   = (0)/,(0) + (I)/(I) + (2)/, (2) + (3)·水3) 42 120 120 54 154 154 141

   The variance of X is given by

10. 

11. 

    (0),(0) + (l)/ (D+(4),(2)+(9),(3)-(밖 141

    

    42 240 360 141 154 154 154 77

    

    321 (141 77 77) 24717 19881 772 77 4836 5929

    (f) The mean of Y is given by

12. 

    = (0)/(0) + (1)左(1)+(2)左(2) 32 30 62

    The variance of Y is given by

    

    62 7.10 32 60 62 7084 3844 77277- 3240 5929

    (g) The covariance of X and Y is given by

13. 

    (I)(I).ULD4(2)(I).U2, l)+(3)(I)/(3,1) 141 62 +2)+2777

    

    12 24 24 36 36 141 62

    

    132 141 62 77 、77)(77 10164 8742 772772 -11422 5929

    (h) The correlation coefficient is given by

14. 

    Cov(X,Y) 1422 772772 772 VL3240人4836 1422 (3240) (4836) 79/12090 24180 =10.359

    (i) The line of the best fit is given by the equation

15. 

    Using the values found in the previous part, we get

16. 

    62 79V1 2090 /3240 141 77 24180 483677 237 215 806 806

    The line of the best fit is shown below