Download 7.4 Mean and Standard Deviation of a Random Variable

7.4 Mean and Standard
Deviation of a Random Variable
Sunday, May 14, 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
• Ex. 7.8 Exam Attempts pg. 369
• Ex. 7.9 Apgar Scores pg. 370
Discrete
• Standard deviation  x
• Ex. 7.10 Defective Components
• Variance:

2
x

x
 bx  gpbg
2
all possible
x values
• Ex. 7.11 Defective Components Revisited
• Ex. 7.12 More on Apgar Scores
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.
• Example pg. 374-375
• Combining random variables - adding sets,
add means. Subtracting sets, subtracting
means – for independent r.v.’s - add variances
• Ex. 7.14 Freeway traffic
• Ex. 7.15 combining exam subscores
• Ex. 7.16 Luggage weights