Chapter 16: Random Variables
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Transcript Chapter 16: Random Variables
Chapter 16: Random Variables
AP Statistics
Vocabulary
• Random Variable: A variable whose value is
based on the outcome of a random event. Use
capital letter, X
• Discrete Random Variable: A random variable
that can take one of a countable number of
distinct outcomes
• Continuous Random Variable: A random
variable that can take any numeric value within a
range of values. The range may be infinite or
bounded at either or both ends.
There is a free game at a carnival that you
attend. In that game, a coin is tossed twice. If
the result is the same outcome (HH or TT) you
win $20. If the result is first a heads and then
a tails, you win $30; but if the result is first a
tails and then a heads, you lose $60. What is
the expect amount you’ll win in the long run?
Expected Value of Random Discrete
Variable
The theoretical long-run average value, the
center of its model. It is a theoretical mean.
E X x Px
There is another game at a the carnival. This
game costs $5 to play. In this game you toss a
die and if a 1, 2, or is rolled, you lose (get no
money); if you roll a 4 or a 5, you win $20 and
if a 6 is rolled, you win $100. What is the
expected amount you will win in the long run?
Obviously, if you play each game for a long
stretch of time, you will win, on average, the
expected value. However, there will be
variation in your winnings. We can compute
this variation.
Variance and Standard Deviation of
Discrete Random Variable
Variance: The expected value of the squared
deviation from the mean or E X
2 Var X x Px
2
Standard Deviation: Describes the spread of
the data.
SD X Var X
Find the variance and standard deviation of the
two games described earlier.
Adding/Subtracting a Constant to
Discrete Random Variable
Suppose that for Game 2, they increase the fee
to play from $5 to $8. What happens to the
expected value, variance and the standard
deviation?
Adding/Subtracting a Constant to
Discrete Random Variable
EX c EX c
Var X c Var X
Multiplying/Dividing a Constant to
Discrete Random Variable
Suppose that the payout for each outcome in
Game 2 is doubled. What happens to the
expected value, variance and standard
deviation?
Multiplying/Dividing a Constant to
Discrete Random Variable
E aX aE X
Var aX a Var X
2
Other “changes”
What if you play the Game 1 twice (or you play
once and your friend plays once)? What is the
expected value, variance and standard
deviation?
Other “changes”
If you play both games, what is the difference in
you expected values, variances, and standard
deviations. We will look at the difference
between Game 1 and Game 2. (Need for the
outcomes of the games to be independent—
or else cannot go forward with problem.)
Other “changes” with discrete (and
continuous) random independent
variables
E X Y E X E Y
Var X Y Var X Var Y
NOTE: You ALWAYS add variances
Continuous Random Variables
Some continuous random variables have normal
models (others do not). All continuous random
models have means and standard deviations (and
variances).
If given the mean and standard deviations of a
continuous random variable that can be modeled
by a normal curve, we can calculate certain
probabilities.
When two independent continuous random
variables have Normal models, so does their sum
or difference.
Example
It has been determined that the annual cost of
medical care for dogs averages $100, with a
standard deviation of $30, and for cats
averages $120, with a standard deviation of
$35.
What is the expected difference in the cost of
medical care for dogs and cats? What is the
standard deviation of that difference?
Example (cont.)
If the difference in cost can be described by a
Normal model, what is the probability that
medical expenses are higher from someone’s
dog than for her cat?
Example (cont.)
What if you have two dogs and a cat? What is
the expected value of your annual medical
costs? Standard deviation?
Example (cont)
What is the probability that your annual medical
bills will exceed $400?