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# Probability : Random Variables, Gamma, Exponential and Lognormal Distributions

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1. Suppose that family incomes in a town are normally distributed with a mean of \$1,200 and a standard deviation of \$600 per month. The probability that a given family has an income over \$2,000 per month is

2. If X is a normal random variable with a mean of 15 and a standard deviation of 3, then P(X = 18) is

3. If telephone calls arrive at a computer telephone service on the average of 30 calls per hour in accordance with a Poisson process, what is the probability that the computer operator will have to wait less than 9.39 minutes for the next nine calls to arrive.

4. A canning company that cans pineapple juice has a machine that automatically fills 16-ounce cans. From quality control data it was found that the average amount of pineapple juice dispensed into the cans was sixteen ounces with a standard deviation of one ounce. If the amount of fill per can is assumed to be normally distributed, what is the probability that the machine will overfill any of the cans?

5. If a random variable X is uniformly distributed over the interval [a, b], then the variance for X is

6. A rock crushing company has three plants, all receiving blasted rocks in bulk. The amount of rocks that can be crushed by one of the plants in one day can be modeled by an exponential distribution. The mean amount of rocks that can be crushed per day by each plant is 4 tons. Assume that the plants operate independently of each other. How many tons of blasted rocks should be stored at any of the plants per day such that the probability of running out of blasted rocks to crush is 0.05?

7. A certain electronic system has a life with an exponential distribution and with mean of 1000 hours. This system is supported by an identical system. The backup system takes over immediately when the system in operation fails. Assume that the systems are operating independently of each other, then the average for the total life of the main and the backup systems is