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Determining Correlation and Distribution

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

(f) Are X and Y independent? Explain
(g) Derive an expression for the round trip time T as a function of X, Y and the speed of the motors
(h) Find Pr{T > 2.5}
(i) Compute E [T]
(j) Do you think X and Y are positively correlated, negatively correlated, or uncorrelated? Why?

Solution Preview

Since all the points are uniformly distributed over the region, the probability is the ratio of the covered area to the total area.
(f) Are X and Y independent? Explain

When all the points are uniformly distributed, the choice of X will not affect the choice of Y, we know that X and Y are independent. The probability of any specific value of X is uncorrelated with the value of Y.
Thus, we can write Pr(X, Y)=Pr(X)* Pr(Y)

(g) ...

Solution Summary

This shows how to determine if variables are independent, create an expression for a given situation, and determine the correlation.