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Economics 671 Problem Set # Multivariate and Conditional Probability
(1) Consider the joint density function
f (x, y) =
{ (^) 2 exp(−[x + y]) for 0 ≤ x ≤ y, 0 ≤ y 0 otherwise.
(a) Show that this is a valid bivariate density function.
(b) Derive the marginal densities for X and Y.
(c) Are the random variables X and Y independent?
(2) Suppose that Corr(X, Y ) = 0. Does this imply that the random variables X and Y are also independent? (Provide a proof or a counter example).
(3) Suppose that an individual is diagnosed with a disease x. This individual then goes on the internet to “research” the disease and finds, in repeated searches, that a far more serious disease y is associated with x. That is, the individual finds (and let’s treat this as a medical certainty) that 80 percent of the individuals who have disease y also have the (less serious) disease x. Our individual then becomes quite concerned that she must also have disease y.
Is this necessarily the case? (Justify your answer in terms of conditional probability. If you believe the concerns are justified, prove it; it you think they are unjustified provide a convincing counter- example).
(4) In a popular game show, a contestant is asked to pick one of three boxes. One of the three boxes is known to contain a prize, while the other two are empty. The game show host knows the location of the prize. After the contestant makes her initial selection, the game show host opens one of the two boxes which was not chosen (and was known by the host to be empty), and shows her that this box is empty. The contestant then has the option to either stay with her original choice, or switch to the other unopened box. If the contestant wishes to maximize her probability of getting the prize, what should she do? (Stay with her original choice, or switch?)
(5) Consider an arbitrary bivariate density function p(x, y). Evaluate whether the following state- ments are true or false in general:
(a) The marginal densities pX (x) and pY (y) are sufficient to define (that is, completely characterize) the joint density p(x, y).
(b) The densities pX (x) and pY |X (y|x) are sufficient to define the joint.
(c) The densities pX (x) and pX|Y (x|y) are sufficient to define the joint.
(d) The densities pX|Y (x|y) and and pY |X (y|x) are sufficient to define the joint.
(Some of these may be obvious, but one of these is certainly not).
