A nut company markets cans of mixed nuts containing almonds, cashews, and peanuts. Suppose the net weight of each can is exactly 1 lb, but the weight contribution of each type of nut is random. Because the three weights sum to 1, a joint probability model for any two gives all necessary information about the weight of the third type. Let $$X$$ be the weight of almonds and let $$Y$$ be the weight of cashews of a randomly selected can. In every can the weight of cashews is at least as much as the weight of almonds.

Suppose the joint pdf of $$X$$ and $$Y$$ is

$f_{X, Y}(x, y) = \begin{cases} 48xy, & 0<x<0.5,\ x<y<1-x,\\ 0, & \text{otherwise.} \end{cases}$

For each of the problems below, set up an integral you would evaluate to solve the problem, and then use software to evaluate the integral. Briefly explain why the integral represents the quantity you’re trying to find; sketching pictures will help. You just have to set up the integrals; you don’t need to evaluate them by hand. You can use software like WolframAlpha or Desmos to do any calculus or create plots, but make sure you clearly set up the expressions to evaluate, and explain your reasoning.

1. Draw a picture representing the region of possible $$(X, Y)$$ values. Explain in context why the possible values are what they are.

2. Explain where the 48 comes from. That is, why is it 48 and not some other value? Justify your answer with an appropriate calculation.

3. Compute $$\text{P}(X > 0.1, Y < 0.4)$$. Interpret the result in context.

4. Compute $$\text{P}(X + Y < 0.7)$$. Interpret the result in context.

5. Find the marginal pdf of $$X$$, and draw a plot of it. Be sure to specify the possible values.

6. Find the marginal cdf of $$X$$. Be sure to define the cdf for any numerical input, regardless of whether the input is a possible value of $$X$$.

7. Compute $$\text{P}(X > 0.1)$$. Interpret the result in context.

8. Find the 75th percentile of $$X$$. Interpret the result in context.

9. Compute $$\text{E}(X)$$. Interpret the result in context.

10. Compute $$\text{SD}(X)$$.

11. Find the marginal pdf of $$Y$$, and draw a plot of it. Be sure to specify the possible values.

12. Find the marginal cdf of $$Y$$. Be sure to define the cdf for any numerical input, regardless of whether the input is a possible value of $$Y$$.

13. Compute $$\text{P}(Y > 0.4)$$. Interpret the result in context.

14. Find the 75th percentile of $$Y$$. Interpret the result in context.

15. Compute $$\text{E}(Y)$$. Interpret the result in context.

16. Compute $$\text{SD}(Y)$$.

17. Compute $$\text{E}(XY)$$.

18. Compute $$\text{Corr}(X, Y)$$.

19. Are $$X$$ and $$Y$$ independent? Explain.