AP STATS
Random Variables
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Expected Value: Center
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A random variable assumes a value based on the outcome of a
random event.
We use a capital letter, like X, to denote a random variable.
A particular value of a random variable will be denoted with a
lower case letter, in this case x.
There are two types of random variables:
Discrete random variables can take one of a finite number of
distinct outcomes.
Example: Number of credit hours
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Continuous random variables can take any numeric value within a
range of values.
• Example: Cost of books this term
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A probability model for a random variable consists of:
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The collection of all possible values of a random variable, and
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The probabilities that the values occur.
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Of particular interest is the value we expect a random variable to
take on, notated μ (for population mean) or E(X) for expected
value.
Example:
You pick a card from a deck. If you get a face card, you win $15. If you
get an ace, you win $25 plus an extra $40 for the ace of hearts. For any
other card you win nothing. Create a probability model for the amount you
win at this game
AP STATS
Random Variables
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
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The expected value of a (discrete) random variable can be found
by summing the products of each possible value by the probability
that it occurs:
Note: Be sure that every possible outcome is included in the sum
and verify that you have a valid probability model to start with.
First Center, Now Spread…
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For data, we calculated the standard deviation by first computing
the deviation from the mean and squaring it. We do that with
random variables as well.
The variance for a random variable is:
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The standard deviation for a random variable is:
More About Means and Variances
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Adding or subtracting a constant from data shifts the mean but
doesn’t change the variance or standard deviation:
Example: Card game
What happens to the expected value and the SD when we add $100 to each
of the winning amounts.
AP STATS
Random Variables
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
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In general, multiplying each value of a random variable by a
constant multiplies the mean by that constant and the variance by
the square of the constant:
Example: Card Game
What happens to the expected value and the SD when we multiply the
winning amounts by 2?
In general,
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The mean of the sum of two random variables is the sum of the
means.
The mean of the difference of two random variables is the
difference of the means.
If the random variables are independent, the variance of their sum
or difference is always the sum of the variances.
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Example: Miguel buys a large bottle and a small bottle of juice. The
amount of juice that the manufacturer puts in the large bottle is a random
variable with a mean of 1024 ml and a standard deviation of 12 ml. The
amount of juice that the manufacturer puts in the small bottle is a random
variable with a mean of 508 ml and a standard deviation of 4 ml. Find the
mean and standard deviation of the total amount of juice in the two bottles.
AP STATS
Random Variables
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Example: Suppose that in one town, 50 year old men have a mean weight
of 177 lb. with a standard deviation of 17 lb. 30 year old men have a mean
weight of 158 lb. with a standard deviation of 12 lb. How much heavier do
you expect a 50 year old man to be than a 30 year old man and what is the
standard deviation of this difference?
Example: A national study found that the average family spent $422 a
month on groceries, with a standard deviation of $84. The average amount
spent on housing (rent or mortgage) was $1120 a month, with standard
deviation $212. What is mean and standard deviation of the total?
Example: In the 4 × 100 relay event, each of four runners runs 100 meters.
A college team is preparing for a competition. The means and standard
deviations of the times (in seconds) of their four runners are as shown in
the table What are the mean and standard deviation of the relay team's
total time in this event? Assume that the runners' performances are
independent.
AP STATS
Random Variables
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Continuous Random Variables
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Random variables that can take on any value in a range of values
are called continuous random variables.
Continuous random variables have means (expected values) and
variances.
We won’t worry about how to calculate these means and variances
in this course, but we can still work with models for continuous
random variables when we’re given the parameters.
Good news: nearly everything we’ve said about how discrete
random variables behave is true of continuous random variables, as
well.
When two independent continuous random variables have Normal
models, so does their
AP STATS
Random Variables
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Example:
Sue buys a large packet of rice. The amount of rice that the manufacturer
puts in the packet is a random variable with a mean of 1016 g and a
standard deviation of 8 g. The amount of rice that Sue uses in a week has a
mean of 210 g and a standard deviation of 40 g. If the weight of the rice
remaining in the packet after a week can be described by a normal model,
what's the probability that the packet still contains more than 895.7 g after
a week?
AP STATS
Random Variables
Name:_____________________________
Date: _____________ Period 1 2 3 4 5 6 7
Bernoulli Trials • The basis for the probability models we will examine in this
chapter is the Bernoulli trial. • We have Bernoulli trials if: • there are two
possible outcomes (success and failure). • the probability of success, p, is
constant. • the trials are independent. Independence • One of the important
requirements for Bernoulli trials is that the trials be independent. • When we
don’t have an infinite population, the trials are not independent. But, there is a
rule that allows us to pretend we have independent trials: • The 10% condition:
Bernoulli trials must be independent. If that assumption is violated, it is still
okay to proceed as long as the sample is smaller than 10% of the population.
The Geometric Model • A single Bernoulli trial is usually not all that interesting.
• A Geometric probability model tells us the probability for a random variable
that counts the number of Bernoulli trials until the first success. • Geometric
models are completely specified by one parameter, p, the probability of success.
Geometric probability model for Bernoulli trials: Geom(p) p = probability of
success q = 1 – p = probability of failure x = # of trials until the first success
occurs PX x       