2.3 Expected Values of Discrete Random Variables
97
While the theorem is far from precise, it does verify that as we move farther away from the mean, in terms of standard deviations, the more of the probability distribution
we cover; hence ¦ is indeed a measure of variability.
Exercises 2.3
1. A small manufacturing firm sells 1 machine per month with probability 0.3; it sells
2 machines per month with probability 0.1; it never sells more than 2 machines per month. If X represents the number of machines sold per month,
a. find the mean and variance of X. b. If the monthly profit is 2 X
2
+ 3X + 1 in thousands of dollars, find the expected monthly profit.
2. Bolts are packaged in boxes so that the mean number of bolts per box is 100
with standard deviation 3. Use Tchebycheff’s Inequality to find a bound on the probability that the box has between 95 and 105 bolts.
3. Graduates of a distinguished undergraduate mathematics program received gradu-
ate school fellowships as follows: 20 received 10,000, 10 received 12,000, 30 received 14,000, 30 received 13,000, 5 received 15,000, and 5
received 17,000. Find the mean and the variance of the value of a graduate fellowship.
4. A fair coin is tossed 4 times; let X denote the number of heads that occur. Find the
mean and variance of X.
5. A batch of 15 electric motors contains 3 defective ones. An inspector chooses 3
without replacement. Find the mean and variance of X, the number of defective motors in the sample.
6. A coin, loaded to show heads with probability
2 3
, is tossed until a head appears or until 5 tosses have been made. Let X denote the number of tosses made. Find the
mean and variance of X.
7. Suppose X is a discrete uniform random variable so that f .x =
1 n
; x = 1; 2; 3; : : : ; n: Find the mean and variance of X.
8. In exercise 5, suppose the batch of motors is accepted if no more than one defective
motor is in the sample. If each motor costs 100 to manufacture, how much should the manufacturer charge for each motor in order to make the expected profit for
the batch be 200?
9. A physicist makes several independent measurements of the specific gravity of a
substance. The limitations of his equipment are such that the standard deviation of each measurement is ¦ units. Suppose ¼ is the true specific gravity of the
98
Chapter 2 Discrete Random Variables and Probability Distributions
Kategori