Exam II Math 411

Boggess

Spring 1996

Clearly show all work using complete sentences. The clarity of your presentation will be graded along with mathematical correctness. Credit will not be given for unsupported answers.

1. Suppose an associate head for a large mathematics department schedules 102 students in a calculus class in a room that only seats 100 students knowing that the probability that a student will drop the class is .05. Use a Poisson distribution to approximate the probability that there will be enough seats for the students that remain in the class.
2. Suppose X is a random variable with density

   

   1. Find c.
   2. Find the distribution function for X.

3. Suppose X is the standard Normal Random Variable, with density . Find the density function for Y=4X+2.
4. Suppose we roll a fair die seven times. What is the expected number of times that a 5 will appear? Derive any formula that you use (i.e. do not just quote a formula for the expected value of this distribution).

5. Suppose X and Y are independent continuous random variables, both with density function

   

   Find the density function of X+Y. You may leave your answer in the form of an integral(s).

6. Suppose X and Y are jointly, uniformly distributed over the region (sketch this region carefully).
   1. Find the joint density function for X and Y.
   2. Find the marginal density functions for X and Y.
   3. Are X and Y independent?

7. A point is chosen at random from the points on the interval and a second point is then chosen at random from the interval between the first point and 1. What is the probability that the distance between the first and second points is at most 1/3? You may express your answer as an integral(s).
8. Suppose the density function of random variable has the following graph. As accurately as possible, sketch the graph of the corresponding distribution function. The area under the left hump is 2/3 and the area under the right hump is 1/3.

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