Let y1 and y2 denote the proportions of two different types of components in a sample from a mixture of chemicals used as an insecticide. suppose that y1 and y2 have the joint density function given by f(y1,y2)= 2, 0≤y1 ≤1,0≤y2 ≤1,0≤y1 +y2 ≤1, 0, elsewhere. (notice that y1 + y2 ≤ 1 because the random variables denote proportions within the same sample.) find a p(y1 ≤ 3/4,y2 ≤ 3/4). b p(y1 ≤ 1/2,y2 ≤ 1/2).

a)

b)

Step-by-step explanation:

For this case we have two random variables Y1 and Y2, the joint density function is given by:

And 0 for other case.

We know that

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

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:

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the answer you are looking for is a..

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step-by-step explanation: