Question: (25 Points.) A company that service air conditioner units in residences and office block is interested in how to schedule its technician in the most efficient manner. If the random variable X , taking the values 1, 2, 3, and 4, is the service time in hours taken at a particular location, and the random variable Y , taking the values 1, 2, 3, and 4, is the number of air conditioner units at the location. The probabilities given by

1. Find joint cumulative distribution function. (2 Points)
2. Find the marginal distribution of X. (1 Point)
3. Find the marginal distribution of Y. (1 Point)
4. Find the expectation of X. (2 Points)
5. Find the expectation of Y. (2 Points)
6. Find the standard deviation of X. (2 Points)
7. Find the standard deviation of Y. (2 Points)
8. Find the conditional probability distribution P(X | Y  4) . (3 Points)
9. Calculate the expectation of P(X | Y = 4). (2 Points)
10. Calculate the standard deviation of P(X | Y= 4). (5 Points)
11. Explain the meaning of differences of two results which are d) and i). (3 Points)

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