Answer:
a) [tex]P(Y_1 \leq 3/4,Y_2 \leq 3/4)=1-2*2* \frac{1/4 *1/4}{2}= 1-\frac{1}{8}=7/8[/tex]
b) [tex]P(Y_1 \leq 1/2,Y_2 \leq 1/2)=1-2*2* \frac{1/2 *1/2}{2}= 1-\frac{1}{2}=1/2[/tex]
Step-by-step explanation:
For this case we have two random variables Y1 and Y2, the joint density function is given by:
[tex] 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[/tex]
And 0 for other case.
We know that [tex]Y_1+Y_2\leq 1[/tex]
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 [tex]y_1 +y_2 \leq 1[/tex]
And we got that [tex] y_2 \leq 1-y_1[/tex] or equivalently in our notation [tex] y \leq 1-x[/tex]. 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:
[tex]P(Y_1 \leq 3/4,Y_2 \leq 3/4)[/tex] we can use the second figure attached.
We see that we have two triangles with the same Area, on this cas [tex]A = \frac{bh}{2}= \frac{1/4 *1/4}{2}[/tex] And then the total area for both triangles is [tex]A_T= 2* \frac{1/4 *1/4}{2}[/tex].
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 :
[tex] V_T = 2*2* \frac{1/4 *1/4}{2}[/tex].
And then we can find the probability like this:
[tex]P(Y_1 \leq 3/4,Y_2 \leq 3/4)=1-2*2* \frac{1/4 *1/4}{2}= 1-\frac{1}{8}=7/8[/tex]
Part b
For this case w want this probability:
[tex]P(Y_1 \leq 1/2,Y_2 \leq 1/2)[/tex] we can use the third figure attached.
We see that we have two triangles with the same Area, on this cas [tex]A = \frac{bh}{2}= \frac{1/2 *1/2}{2}[/tex] And then the total area for both triangles is [tex]A_T= 2* \frac{1/2 *1/2}{2}[/tex].
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 :
[tex] V_T = 2*2*\frac{1/2 *1/2}{2}[/tex].
And then we can find the probability like this:
[tex]P(Y_1 \leq 1/2,Y_2 \leq 1/2)=1-2*2* \frac{1/2 *1/2}{2}= 1-\frac{1}{2}=1/2[/tex]
