Joint/Marginal Probability Distributions :)?

Question

Thanks :)

1. The joint density for x and y is given by f(x,y) = 1/n2, x = 1, 2, 3, ..n, y = 1, 2, 3, ..n where n is a positive integer
a. Verify that f(x,y) satisfies the conditions necessary to be a density
b. Find the marginal densities for x & y
c. Are x & y independent?

3. Let X denote the temperature (oC) and let Y the time in minutes that it take for the diesel engine on an automobile to get ready to start. Assume that the joint density for x & y is given by f(x,y) = c(4x+2y+1), 0 ≤ x ≤ 40, 0 ≤ y ≤ 2
a. Find the value of c
b. Find the probability that on a randomly selected day, the air temperature will exceed 20oC and it will take at least 1 minute for the car to be ready to start
c. Find the marginal densities for x & y
d. Are x & y independent?
e. Find the probability that in a randomly selected day it will take at least 1 minute for the car to be ready to start

Thanks again! :)

Answer 1

I'll try to help you with (3). I got confused about (1) and (2), maybe you can clarify later.

(3)
(a) The integral of the joint distribution over its domain must be 1. So, Int(x =0 to 40)Int(y = 0 to 2) c(4x + 2y + 1)dx dy = c Int (y = 0 to 2) [2x^2 + 2xy + x] (x = 0 to 40) dy = c Int (0 to 2) (80 y + 3240) dy = 40 c Int (2y + 81) dy = 40c [y^2 + 81y] (0 to 2) = 40c [4 + 162] =1 => c = 1/(40 * 166) = 0.000150602


(b) It's the probability that X > 20 and Y >=1. So, P(X>20, Y>1) = 0.000150602 Int (4x + 2y + 1) over the rectangle {20 < x <=40 , 1 <= y <=2}. Just algebra

(c) The marginal distribution Fax for x is fx(x) = Int (y =0 to 2) 0.000150602 (4x+2y+1) Dy = 0.000150602 [4xy + y^2 + y] (0 to 2) =0.000150602(8x + 6) = 0.000301205 (4x + 3)
For y, we have fy(y) = Int (x =0 to 40) 0.000150602 (4x+2y+1) dx = 0.000150602 [2x^2 + 2yx + x] (x= 0 to 40) = 0.000150602 (80y + 3240) = 0.006024096 (2y + 81).

d)No, because their joint distribution is not the product of their marginal distributions.

(e) We have to find P(Y >=1). Since we already have fx, we integrate it from 1 to 2.
0.006024096 Int (2y + 81)dy (1 to 2) = 0.006024096 (y^2 + 81y) (from 1 to 2). We plug in 2 and 1 and subtract.