514 Suppose that the random variables Y1 and Y2 have joint p

5.14 Suppose that the random variables Y1 and Y2 have joint probability density function f(y1, y2) given by a Verify that this is a valid joint density function. b What is the probability that Y1 + Y2 is less than 1?

Solution

Let y1 be the x-axis and y2 be the y axis on a plot. We are going to be in the first quadrant and will have two lines. y = x and y = 2-x, these come from the inequalities.
The area we are interested in is bounded by the y-axis, y =x and y = 2-x, the triangle right against the y-axis. The next step to verify that f is a probability density function is to integrate it over this area. To do this you\'ll have to split the double integral into to parts

f(y1,y2) dy1 dy2 + f(y1,y2) dy1 dy2

the first double integral has limits of: 0 < y1 < y2, 0 < y2 < 1
the second double integral has limits of: 0 < y1 < 2 - y2, 1 < y2 < 2

the first double integral equates to 2/5
the second double integral equates to 3/5

the sum is 1, and we have verified f_x is a pdf.

you can also do this a one double integral

f (y1, y2) dy2 dy1

with limits

y1 < y2 < 2 - y1
0 < y1 < 1

2. Again we need to draw the area & integrate over:

Double integral

f (y1, y2) dy2 dy1

with limits:

y1 < y2 < 1 - y1
0 < y1 < 1/2

and the solution is: 1/32