Miscellaneous

Step 1: The Poisson distribution formula is: where is the average rate of occurrence, is the number of occurrences, and is the base of the natural logarithm.
Step 2: Mean expectation ( ): The average number of floods in 10 years is .
Step 3: Probability of 3 or fewer overflows ( ): Using the Poisson formula: - For : - For : - For : - For :
Step 4: Add the probabilities: Substitute the values:
Final Answers: - Mean expectation: . - Probability of 3 or fewer overflows: or approximately .

Step 1: For a binomial distribution, the mean is given by: Since Poisson approximation is used, the mean of the Poisson distribution is also 8. Thus, Assertion (A) is true.
Step 2: The probability mass function of a Poisson distribution is: Substituting and : Since the given expression matches this calculation, Reason (R) is also true.
Step 3: However, Reason (R) does not directly explain why the mean of the Poisson distribution is 8. The mean of a Poisson distribution is derived from the binomial approximation ( ), not from the probability calculation of . Thus, Assertion (A) and Reason (R) are both true, but Reason (R) is not the correct explanation of Assertion (A).

Step 1: Divide 1275 hours by 24 to find the remainder:
Step 2: The remainder 3 means the time advances by 3 hours from 9:00 p.m., which is 9:00 p.m. the next day.

Step 1: Assign coded time variables ( ) for convenience:
Let for the years 1996 to 2001, respectively.

Step 2: Tabulate the data:
 

Step 3: Use the least squares formula for straight-line trend:

where:

Substitute the values:

Step 4: Write the equation:

Step 1: The objective function is maximized at one of the vertices of the feasible region . The vertices of are , , and .
Step 2: Evaluate at each vertex: - At : - At : - At :
Step 3: Conclusion: The maximum value of is 450 at .

Trivial constraints correspond to the boundaries of the graph in the first quadrant.
- Since the graph is in the first quadrant, and are the two trivial constraints.