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Ch15 Discrete Random Variables
Name: _____________________________
RANDOM VARIABLE – a variable whose value is a numerical outcome of a random phenomenon.
Two type of Random Variables:
DISCRETE – countable number of possible values
CONTINUOUS – values in an interval of numbers (observations that are measured)
Example 1: Discrete or Continuous?
a. Number of defective tires on a car
b. Volume of water in Lake Travis
c. Number of student’s on Friday’s absence list
d. Home team score in a basketball game
e. Heights of females at Clements HS
f. Time needed to walk to your next class
Discrete Distributions
Probability Distributions (models) for Discrete Random Variable X:
Value of X
x1
x2
x3
….
xk
Probability
p1
p2
p3
….
pk
2*** p1 + p2
+ p3
Properties of Discrete Random Variables:
1*** 0  p ( x )  1
+ ….
+ pk = 1

p( x)  1
all x values
3*** The Mean of a discrete random variable, denoted by

 x , is found by:  x 
x  p( x)
all x values
(Note the term expected value is sometimes used in place of mean value, and E(x) is an alternative notation for
4*** The Variance of a discrete random variable x, denoted by

2
 x2 , is found by:  x 
x  
2
x )
 p( x)
all possible x values
5*** The standard deviation of x, denoted by
 x , is the square root of the variance.
Example 2: Let X be the number of daughters in a family with 3 children.
a. Write the probability distribution of X.
b. Find the probability of having more than 1 girl.
c. Find the mean number of daughters in this family.
Answers to practice problems:
1) b. $23.61 c. $38.16 d. $23 or less
5) a. 0.949 b.  x  3.315 defects c.  x
d. Find the standard deviation of the number of daughters.
2) b. 1.75 children d. .875 boys
 1.65 defects
3) $27,000
4) $10,600
Example 3: Nonstandard dice can produce interesting distributions of outcomes. You have two balanced, six-sided dice. One is a
standard die, with faces having 1, 2, 3, 4, 5, and 6 spots. The other die has three faces with 0 spots and three faces with 6 spots. Find the
probability distribution for the total number of spots Y on the up-faces when you roll these two dice.
Example 4:
A discrete random variable X can assume five possible values: 1, 2, 3, 4, 5. Its probability distribution is shown here:
X
1
2
3
4
5
P(X)
.2
.1
?
.25
.15
a. Find the probability that X is less than or equal to 3?
b. Find the probability that X is more than 2 but less than or equal to 5)?
c. determine the expected value and standard deviation.
d. Find the probability that X is within one standard deviation of the mean.
Practice: Work on a separate sheet of paper!
1. You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6 on the second time, you win $50. If not, you
lose.
a. Create a probability distribution for the amount you win at this game
b. Find the expected amount you’ll win
c. Find the standard deviation of the amount you might win
d. How much would you be willing to pay to play this game? Explain.
2. A couple plans to have children until they have a girl, but they agree that they will not have more than three children even if they are all
boys.
a. Create a probability distribution for the number of children they’ll have
b. Find the expected number of children
c. Create a probability distribution for the number of boys they’ll have
d. Find the expected number of boys they’ll have
3. A small software company bids on two contracts. It anticipates a profit of $50,000 if it gets the larger contract and a profit of $20,000
on the smaller contract. The company estimates there is a 30% chance it will get the larger contract and a 60% chance it will get the
smaller contract. Assuming the contracts will be awarded independently, what’s the expected profit? (Hint: create a probability model)
4. A man buys a racehorse for $20,000 and enters it in two races. He plans to sell the horse afterward, hoping to make a profit. If the horse
wins both races its value will jump to $100,000. If it wins one of the races, it will be worth $50,000. If it loses both races, it will be worth
only $10,000. The man believes there’s a 20% chance that the horse will win the first race and a 30% chance it will win the second one.
Assuming that the two races are independent events, find the man’s expected profit. (Hint: create a probability model)
5. A consumer organization that evaluates new automobiles customarily reports the number of major defects on each car examined. Let X
denote the number of major defects on a randomly selected car of a certain type. A large number of automobiles were evaluated, and a
probability distribution consistent with these observations is
X
0
1
2
3
4
5
6
7
8
9
10
P(X)
.041
.010
.329
.223
.178
.114
.061
.028
.011
.004
a. Find the probability that the number of defects is at least 2.
b. Find the expected number of defects and the standard deviation.
c. Find the probability that X is within one standard deviation of the mean.