Answer:
a)
b)
Step-by-step explanation:
For this case we have two random variables Y1 and Y2, the joint density function is given by:
![f(y_1,y_2) = 2, \leq y_1 \leq 1, 0\leq y_2 \leq y_2, 0 \leq y_1 +y_2 \leq 1](https://tex.z-dn.net/?f=%20f%28y_1%2Cy_2%29%20%3D%202%2C%20%5Cleq%20y_1%20%5Cleq%201%2C%200%5Cleq%20y_2%20%5Cleq%20y_2%2C%200%20%5Cleq%20y_1%20%2By_2%20%5Cleq%201)
And 0 for other case.
We know that ![Y_1+Y_2\leq 1](https://tex.z-dn.net/?f=Y_1%2BY_2%5Cleq%201)
Let Y1 =X and Y2 =Y we can plot the joint density function. First we need to solve the slope line equation from the condition ![y_1 +y_2 \leq 1](https://tex.z-dn.net/?f=y_1%20%2By_2%20%5Cleq%201)
And we got that
or equivalently in our notation
. And we know that the two random variables are between 0 and 1. So then the joint density plot would be given on the figure attached.
Part a
In order to find the probability that:
we can use the second figure attached.
We see that we have two triangles with the same Area, on this cas
And then the total area for both triangles is
.
Since our density function have a height of 2 since the joint density is equal to 2 then we can find the volume for the two triangles like this :
.
And then we can find the probability like this:
Part b
For this case w want this probability:
we can use the third figure attached.
We see that we have two triangles with the same Area, on this cas
And then the total area for both triangles is
.
Since our density function have a height of 2 since the joint density is equal to 2 then we can find the volume for the two triangles like this :
.
And then we can find the probability like this: