In this chapter we introduce the idea of a
random variable as well as looking at its
shape, center, and spread.
A numerical variable whose value is based on the
outcome of a random event is called a
.
If we can list the possible values of the variable it is
called
, otherwise it is called
.
?Examples?
The collection of all possible values and the
probabilities of each occurring is called the
of the variable.
Let x = the value rolled on a standard 6-sided die. The
probability model of x is:
x
P(x)
1
1
6
» 0.1667
2
1
6
» 0.1667
3
1
6
» 0.1667
4
1
6
» 0.1667
5
1
6
» 0.1667
6
1
6
» 0.1667
You roll a 6-sided die. If it comes up 5 you win $100.
If not you roll again. If it comes up 5, you win $50. If
not, you pay $20. Let x = amount you “win” when
you play this game once. Give the probability model
of x.
The
or
of a random variable x
represents the mean of the numerical data set that
could be created by observing the value of a random
variable many, many, many times (an infinite number
of times). This is denoted m x , m ( x ), or E ( x ) .
It is calculated by:
mx = å x × P ( x)
The
of a random variable
represents the standard deviation of the numerical data
set described above. This is denoted s x or s ( x ).
It is calculated by:
sx =
å( x - m x ) × P ( x )
2
You roll a 6-sided die. If it comes up 5 you win $100.
If not you roll again. If it comes up 5, you win $50. If
not, you pay $20. Let x = amount you “win” when
you play this game once. Find the mean and the
standard deviation of x.
Both the mean and the standard deviation of a random
variable can be calculated using our TI 83/84.
Put the values of x in L1 and the corresponding
probabilities in L2. Then:
Here is the calculator screen
for the previous example:
A carnival game offers a $100 cash prize for anyone
who can break a balloon by throwing a dart at it from a
certain distance. It costs $5 to play, and you’re willing
to spend up to $20 trying to win, but if you win before
spending $20, you will stop. Let x = the amount you
profit from this game, and suppose there is a 10%
chance of breaking the balloon on any one throw.
Give the distribution of x.