EXERCISES Section 3.1 (1-10)

1

A concrete beam may fail either by shear $\left(S\right)$ or flexure $\left(F\right)$. Suppose that three failed beams are randomly selected and the type of failure is determined for each one. Let $X=$ the number of beams among the three selected that failed by shear. List each outcome in the sample space along with the associated value of $X$.

2

Give three examples of Bernoulli rv’s (other than those in the text).

3

Using the experiment in Example 3.3, define two more random variables and list the possible values of each.

4

Let $X=$ the number of nonzero digits in a randomly selected 4-digit PIN that has no restriction on the digits.

What are the possible values of $X$ ? Give three possible outcomes and their associated $X$ values.

5

If the sample space $\mathcal{S}$ is an infinite set, does this necessarily imply that any rv $X$ defined from $\mathcal{S}$ will have an infinite set of possible values? If yes, say why. If no, give an example.

6

Starting at a fixed time, each car entering an intersection is observed to see whether it turns left $\left(L\right)$ , right $\left(R\right)$ , or goes straight ahead $\left(A\right)$. The experiment terminates as soon as a car is observed to turn left. Let $X=$ the number of cars observed. What are possible $X$ values? List five outcomes and their associated $X$ values.

7

For each random variable defined here, describe the set of possible values for the variable, and state whether the variable is discrete.

a. $X=$ the number of unbroken eggs in a randomly chosen standard egg carton

b. $Y=$ the number of students on a class list for a particular course who are absent on the first day of classes

c. $U=$ the number of times a duffer has to swing at a golf ball before hitting it

d. $X=$ the length of a randomly selected rattlesnake

e. $Z=$ the sales tax percentage for a randomly selected amazon.com purchase

f. $Y=$ the $\mathrm{pH}$ of a randomly chosen soil sample

g. $X=$ the tension (psi) at which a randomly selected tennis racket has been strung

h. $X=$ the total number of times three tennis players must spin their rackets to obtain something other than $UUU$ or $DDD$ (to determine which two play next)

8

Each time a component is tested, the trial is a success $\left(S\right)$ or failure $\left(F\right)$. Suppose the component is tested repeatedly until a success occurs on three consecutive trials. Let $Y$ denote the number of trials necessary to achieve this. List all outcomes corresponding to the five smallest possible values of $Y$ , and state which $Y$ value is associated with each one.

9

An individual named Claudius is located at the point 0 in the accompanying diagram.

Using an appropriate randomization device (such as a tetrahedral die, one having four sides), Claudius first moves to one of the four locations $B_1,B_2,B_3,B_4$. Once at one of these locations, another randomization device is used to decide whether Claudius next returns to 0 or next visits one of the other two adjacent points. This process then continues; after each move, another move to one of the (new) adjacent points is determined by tossing an appropriate die or coin.

a. Let $X=$ the number of moves that Claudius makes before first returning to 0. What are possible values of $X$? Is $X$ discrete or continuous?

b. If moves are allowed also along the diagonal paths connecting 0 to $A_1,A_2,A_3$ , and $A_4$ , respectively, answer the questions in part (a).

10

The number of pumps in use at both a six-pump station and a four-pump station will be determined. Give the possible values for each of the following random variables:

a. $T=$ the total number of pumps in use

b. $X=$ the difference between the numbers in use at stations 1 and 2

c. $U=$ the maximum number of pumps in use at either station

d. $Z=$ the number of stations having exactly two pumps in use