Econ 310, Spring 2014- Week 6:
Problem 1- The weekly output of a steel mill is a uniformly distributed random variable that lies between 110 and 175 metric tons.
1. Compute the probability that the steel mill will produce more than 150 metric tons next week.
2. Determine the probability that the steel mill will produce between 120 and 160 metric tons next week.
3. Compute the expectation and variance of the weekly output from the steel mill.
Problem 2- The following density function describes the random variable X
1. Graph the density function.
2. Find the probability that X lies between 1 and 3.
3. Find the probability that X lies between 4 and 8.
Problem 3- (Mutually Exclusive implies Dependence) Given any set A and B such that P(A) > 0, P(B) > 0, and A ∩ B = ∅. Please show that A and B are not independent sets.
Problem 4- (Conditional Probability and Expectation) Consider following 2 × 2 joint probability table:
|
|
X=1
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X=2
|
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Y=1
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0.3
|
0.1
|
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Y=2
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0.1
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0.5
|
1. Please compute the probability P(X = 1) and P(Y = 2|X = 1).
2. Please compute E(X) and E(Y|X = 1).
Problem 5- Given two random variable X and Y , suppose E(X) = 4, E(Y ) = 8, V (X) = 4, V (Y ) = 9 and Cov(X, Y ) = 0.5.
1. What's the correlation between X and Y?
2. Define Z = (3/4)X + (1/4)Y. Please compute E(Z) and V (Z).