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Chapter 16: Random Variables
Random Variables
A numerical variable whose value is based on the outcome of a random event is called a
random variable
•
if we can list the possible values of the variable it is called discrete, otherwise it is
called continuous
• x = value rolled on a die is discrete
• x = how long it takes to drive to work on any given day is continuous
•
The collection of all possible values and the probabilities of each occurring is called
the probability model of the variable
• x = the value rolled on a standard 6-sided die
x
P(x)
1
1
2
1
3
1
4
1
5
1
6
1
/6 ≈ 0.1667
/6 ≈ 0.1667
/6 ≈ 0.1667
/6 ≈ 0.1667
/6 ≈ 0.1667
/6 ≈ 0.1667
Example 1: Carnival Game #1
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.
Def.
The mean or expected value 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 ) .
The standard deviation of a random variable represents the standard deviation of the
numerical data set described above. This is denoted s x or s ( x ) .
•
mx = å x × P ( x)
•
sx =
å( x - m )
2
x
× P ( x)
Example 2: Carnival Game #1 Revisited
Find the m x and s x for the random variable described in Example 1.
This can also be done via the TI calculator. Put the values of x in L1 and the corresponding
probabilities in L2, and the run the 1-Var Stats command with L1, L2 next to it:
Press
Press
then enter the values in L1 and L2
Example 3: Carnival Game #2
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.
Now work on Random Variables Practice handout