The joint density of Y1 , the proportion of the capacity of the tank that is stocked at the beginning of the week, and Y2 , the proportion of the capacity sold during the week, is given by
f(y1,y2)={
3*y1, if 0 ≤ y2 ≤ y1 ≤ 1,
0, elsewhere
}
Question: Find P(Y2 ≤ Y1/2)
, the probability that the amount sold is less than half the amount purchased.
I am not sure how to set up the integral for the limits Y2 ≤ Y1/2
in P(Y2 ≤ Y1/2)
. I know that 3*y1
is bounded between 0 ≤ y2 ≤ y1 ≤ 1 so that forms a triangle beneath the line y = x with base and height equal to 1.
But for Y2 ≤ Y1/2
do I draw a separate graph, or do I bound the Y2 ≤ Y1/2
into that same graph of 0 ≤ y2 ≤ y1 ≤ 1 which is given in the original question.
I just need a starting point. I am not sure where to begin in this question.
Best Answer
Start with one variable, $y_1$, ignoring the restrictions from $y_2$ examine the restrictions on the variable $y_1$. We have $0\leq y_1 \leq 1$.
Now that you have details of $y_1$ you can examine $y_2$ in terms of $y_1$. Given the value of $y_1$ what are the restrictions on $y_2$? We have $0 \leq y_2 \leq y_1$ from the first part of the question and $y_2 \leq \frac{y_1}{2}$ from the second part of the question. Combining these two requirements on $y_2$ gives $0 \leq y_2 \leq \frac{y_1}{2}$
Therefore the integral is
$\int_{y_1=0}^{1} \left(\int_{y_2=0}^{\frac{y_1}{2}} f(y_1,y_2)dy_2 \right) dy_1$
Notice that the limits of $y_2$ contain information about $y_1$ so the integration with respect to $y_2$ has to be done first.