Probability density function
6) Suppose we have a building with a floor shaped like an isosceles right triangle. The two sides adjacent to the right triangle have length 100 feet. Think of the right angle being at the origin, and other two corners at (100, 0) and (0, 100). The overhead crane is located at the origin and needs to travel to a point (X, Y), which is uniformly distributed over the region. The crane has two motors one that moves the crane north and south, and the other that moves the crane east and west. Both motors move at the speed of 20 feet per minute. Since the motors can work at the same time, it is reasonable to assume that the length of time to go from the origin to (X, Y) is the maximum of two times: the time to go east from 0 to X, and the time to go north from 0 to Y. Let T be the length of time that it takes the crane to move from origin to (X, Y) and return to the origin
(a) What is the joint probability density function of (X, Y)?
(b) Find the marginal probability density function of X
(c) Use symmetry to determine the marginal probability density function of Y without integrating
(d) What is the conditional probability density function of Y given X = 10?
(e) What is the E [Y | X =10]?
This shows how to compute joint, conditional, and marginal probability density functions.
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