Transcript 7.4 Mean and Standard Deviation of a Random Variable

7.4 Mean and Standard
Deviation of a Random Variable
Saturday, April 8, 2017
Vocabulary
• Mean value,  x , describes where the probability
distribution is centered.
• Standard deviation,  x , describes variability in the
probability distribution. When  xis small (little
variability) values of x tend to be close to  x and
when  xis large (more variability) values of x tend
to be farther away from  x
• (Remember x and s are for a sample and  x
and  x are for a population and in a probability
distribution we know all possible outcomes.
“mean of the random variable x” and “mean of
the probability distribution of x” are
interchangeable/mean the same thing
“Standard deviation of the probability
distribution of x” and “ std. dev. of the random
variable x” are interchangeable as well.
Look at figure 7.10 on page 367
Discrete
• Mean value (expected value):  x
x 
x px
 bgbg
all possible
x values
Discrete
• Standard deviation  x
• Variance:

2
x

x
 bx  gpbg
2
all possible
x values
Example 7.8/7.9
Individuals applying for a
certain license are
allowed up to four
attempts to pass the
licensing exam. Let x
denote the number of
attempts made by a
randomly selected
applicant. The
probability distribution
is as followed:
x
1
2
3
4
p(x) .1
.2
.3
.4
Find the expected
value, variance, and
standard deviation of
the distribution.
Continuous
•  x and  x are defined and calculated using
calculus we won’t be calculating them only
interpreting them and they have the same
meanings as in the discrete case.
• Ex. 7.13
Mean and Variance of Linear Functions
and Linear Combinations (rules for
means and variances)
• What happens to a mean when we add to a data set?
Add same to the mean
• What happens to a mean when we multiply a data set
by some factor? Multiple same to mean
• What happens to standard deviation when we add to a
data set? Nothing – stays the same
• What happens to standard deviation when we multiply
a data set by some factor? Multiply the variance by
the square of the factor and then square root.
*always work with the variance and then square root
back.
• Combining random variables - adding sets,
add means. Subtracting sets, subtracting
means – for independent r.v.’s - add variances
– don’t add standard deviations. Always add
don’t subtract variances even when you are
subtracting
• Ex. 7.14 Freeway traffic
• Ex. 7.15 combining exam subscores
• Ex. 7.16 Luggage weights
Example
You’re planning to spend next year wandering through
the mountains of Kyrgyzstan. You plan to sell your
used SUV so you can purchase an off-road Honda
motor scooter when you get there. Used SUV’s are
selling for a mean of $6940 and standard deviation of
$250. Your research shows that scooters in Kyrgyzstan
are going for about 65,000 som with a standard
deviation of 500 som. One U.S. dollar is worth about
38.5 Kyrgyzstan som.
How much cash can you expect to pocket after you sell
your SUV and buy the scooter? Find the variance and
standard deviation as well.