Question

In: Statistics and Probability

18. Joint Probability Calculations: a) What is a joint probability density function? i. What is an...

18. Joint Probability Calculations:

a) What is a joint probability density function?

i. What is an example of a discrete joint probability function?

ii. What is an example of a continuous joint probability function?

b) What are marginal density/mass distribution functions? How can we calculate them from the joint density/mass distribution functions?

i. Can you calculate the marginals when the joint space is not a rectangle? (e.g. The space of jpdf[x,y] is 0 < x < y, 0 < y < 1)

c) How do you know that X and Y are independent if Freq[{x,y},JointXY] = Freq[x,X] Freq[y,Y] or jpdfX,Y[x, y] = pdfX[x] pdfY[y] ?

i. Why is it that independent random variables have rectangular joint spaces but a rectangular joint space does not always imply two independent variables?

d) How do you calculate probabilities with joint probability density functions, both continuous and discrete?

e) How is correlation[X, Y] defined, for both continuous and discrete random variables?

i. How do we interpret the value produced by this formula?

ii. Why is it that independent random variables always have a correlation of zero, but a correlation of zero does not imply two independent variables?

f) How do you calculate conditional probabilities for both continuous and discrete random variables?

i. Can you calculate conditional expectations for both continuous and random variables?

Expert Solution

orchestra answered 2 years ago

. The joint probability density function of X and Y is given by ?(?, ?) =...

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