Transcript Chapter 7

Chapter 7
Random Variables
7.1: Discrete and Continuous Random
Variables
Random Variables
• A random variable is a variable whose
value is a numerical outcome of a random
phenomenon.
– the basic units of sampling distributions.
– 2 types: discrete and continuous
Discrete Random Variables
• A discrete random variable X has a countable
number of possible values.
• The probability distribution of a discrete random
variable X lists the values and their
probabilities.
Value of X: x1 x2 x3 … xk
Probability: p1 p2 p3 … pk
Probability Distribution
• The probability pi must satisfy two
requirements.
– Every probability pi is an number between 0
and 1
– The sum of the probabilities is 1:
p1 + p2 + p3 +…+pk = 1
• Find the probability of any event by adding
the probabilities pi of the particular values
xi that make up the event.
Example: Maturation of male
college students
• In an article in the journal Developmental
Psychology (March 1986), a probability
distribution for the age X (in years) when
male college students began to shave
regularly is shown:
X
11
12
13
14
15
P(X)
0.013
0
0.027
0.067
0.213
X
16
17
18
19
≥20
P(X)
0.267
0.240
0.093
0.067
0.013
Example
• Page 470 #7.2
Example
• Page 470 #7.4
Continuous Random Variables
• A continuous random variable X takes on
all values in an interval of numbers.
• The probability distribution of X is
described by a density curve. The
probability of any event is the area under
the density curve and above the values of X
that make up that event.
• All continuous probability distributions
assign probability 0 to every individual
outcome.
Example: Violence in Schools
• Page 476 #7.9
Example: Drugs in schools
• An opinion poll asks a SRS of 1500
American adults what they consider to be
the most serious problem facing our
schools. Suppose that if we could ask all
adults this question, 30% would say
“drugs”. What is the probability that the
poll result differs from the truth about the
population by more than two percentage
points? N(.3, 0.0118)
Chapter 7
Random Variables
7.2: Means and Variances of Random
Variables
Activity 7B
• Page 481
Mean and expected Value
• Mean of a probability distribution is
denoted by µ, or µx.
• The mean of the random variable, X is
often referred to as the expected value of X.
Mean of a
Discrete Random Variable
• Suppose that X is a discrete random variable
whose distribution is
Value of X: x1 x2 x3 … xk
Probability: p1 p2 p3 … pk
To find the mean of X, multiply each possible value
by its probability, then add all the products.
Example
• Page 486 #7.24
Example
• Using the data from the “Maturation of
male college students” example, find and
interpret the mean.
Variance of a
Discrete Random Variable
• Suppose that X is a discrete random variable
whose distribution is
Value of X: x1 x2 x3 … xk
Probability: p1 p2 p3 … pk
and that µ is the mean of X. The variance of X is
σx2 = Σ(x1 - µx)2pi.
The standard deviation σx of X is the square root of
the variance.
Example
• Page 486 #7.28
Technology Tip
• To find µx and σx:
Example
• Using the data from the “Maturation of
male college students” example, find the
standard deviation.
• Use the empirical rule to determine if the
“Maturation of male college students” data
is normally distributed.
Sampling Distributions
• The sampling distributions of statistics are
just the probability distributions of these
random variables.
Law of Large Numbers
• The average of a randomly selected sample
from a large population is likely to be close
to the average of the whole population.
Law of Large Numbers
• What is the mean of rolling 3 dice?
Example:
Emergency Evacuations
• A panel of meteorological and civil engineers
studying emergency evacuation plans for
Florida’s Gulf Coast in the event of a hurricane
has estimated that it take between 13 and 18 hours
to evacuate people living in low-lying land, with
the probabilities shown in the table.
Let X = the time it takes a randomly selected person
living in low-lying land in Florida to evacuate.
Time to Evacuate Probability
(nearest hour)
13
0.04
14
0.25
15
0.40
16
0.18
17
0.10
18
0.03
• Is X a discrete random variable or a
continuous random variable?
• Sketch a probability histogram for this data.
• Find and interpret the mean.
• Find the standard deviation.
• Weather forecasters say that they cannot
accurately predict a hurricane landfall more
than 14 hours in advance.
Find the probability that all residents of
low-lying areas are evacuated safely if the
Gulf Coast Civil Engineering Department
waits until the 14-hour warning before
beginning evacuation.
Law of Small Numbers
• Gambler’s Fallacy is the belief that every
segment of a random sequence should
reflect the true proportion.
• This is a myth. There is no law of small
numbers!
Rules for Means
• Rule 1: If X is a random variable and a and
b are fixed numbers, then
μa+bx = a + bμx
• Rule 2: If X and Y are random variables,
then
μX + Y = μX + μY
Rules for Variances
• Rule 1: If X is a random variable and a and
b are fixed numbers, then
σ2a+bx = b2σ2x
• Rule 2: If X and Y are independent random
variables, then
σ2X + Y = σ2x + σ2Y
σ2X - Y = σ2x + σ2Y
Means and Variances
• Variances add, standard deviations don’t.
• These rules can extend for more than 2
random variables…just follow the pattern.
Combining Normal Random
Variables
• Any linear combination of independent
Normal random variables is also Normally
distributed.
Example
• Page 499 #7.38
• Page 499 #7.40
• Page 500 #7.42
Example
• Page 501 #7.44
Example
• Page 503 #7.50
Example
• Page 503 #7.52