Transcript p.p chapter 6.1

Discrete and Continuous
Random Variables
Section 6.1
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
Objectives
1. Discrete Random Variables
1. What is a discrete random variable?
2. Mean (Expected Value) of a DRV
1.
Examples: Apgar Scores of Babies, Roulette
3. Standard Deviation (and variance) of a DRV
1.
Calculator saves the day!
2. Continuous Random Variables
1. What is a continuous random variable?
1.
Area under the curve!
2. Finding the probability of the interval of outcomes,
Z-scores return!
Intro Example:
• Suppose we toss a fair coin 3 times. The sample
space for this chance process is:
• HHH HHT HTH THH HTT THT TTH TTT
• Since there are 8 equally likely outcomes the probability
is 1/8 for each possible outcome.
• Define the variable X = the number of heads obtained.
• What are my outcomes of possible heads?
X=0
X=1
X=2
X=3
TTT
HTT, THT, TTH
HHT, HTH, THH
HHH
Intro Example
• We can summarize the probability
distribution of X as follows:
Value (X) :
0
1
2
3
Probability:
1/8
3/8
3/8
1/8
We just talked about a discrete random variable!
- A discrete random variable X takes a fixed set
of possible values with gaps between
What is A Discrete Random
Variable
• We have learned several rules of probability but
one way of assigning probabilities to events:
assign probabilities to every individual outcome,
then add these probabilities to find the
probability of any event.
• This idea works well if we can find a way to list
all possible outcomes. We will call random
variables having probability assigned in this way
discrete random variables.
Value:
…..
Probability:
…..
Requirements of DRV
Apgar Scores: Babies’ Health at Birth
• In 1952, Dr. Virginia Apgar suggested five criteria for
measuring a baby’s health at birth: skin color, heart rate,
muscle tone, breathing, and response when stimulated.
She developed a 0-1-2 scale to rate a newborn on each
of the five criteria. A baby’s Apgar score is the sum of
the ratings on each of the five scales, which gives a
whole-number value from 0 to 10. Apgar scores are still
used today to evaluate the health of newborns.
Apgar Scores: Babies’ Health at Birth
• What Apgar scores are typical? To find out, researchers
recorded the Apgar scores of over 2 million newborn
babies in a single year. Imagine selecting one of these
newborns at random. (that’s our chance process). Define
the random variable X = Apgar score of a randomly
selected baby one minute after birth. The table below gives
the probability distribution for X.
Value
0
Probability: .001
1
2
3
4
5
6
7
8
9
10
.006 .007 .008 .012 .020 .038 .099 .319 .437 .053
Apgar Scores: Babies’ Health at Birth
A) Show that the probability distribution for X is
legitimate
B) Doctors decided that Apgar scores of 7 or
higher indicate of healthy baby. What's the
probability that a randomly selected baby is
healthy.
Mean (Expected Value) Of A
Discrete Random Variable
Winning (and losing) at Roulette
• On an American roulette wheel, there are 38 slots numbered 1
through 36, plus 0 and 00. Half of the slots from 1 to 36 are red; the
other half are black. Both the 0 and 00 slots are green. Suppose that a
player places a simple $1 bet on red. If the ball lands on a red slot, the
player gets the original dollar back, plus an additional dollar for
winning the bet. If the ball lands in a different-colored slot, the player
loses the dollar bet to the casino.
• Lets define the random variable X = net gain from a single $1 bet on
red. The possible values of X are -$1 and $1 (the player either gains a
dollar or loses a dollar.) What are the corresponding probabilities? The
chance that the ball lands on red slot is 18/38. The chance that the
ball lands in a different-colored slot is 20/38. Here is the probability
distribution of X:
Value
-$1
$1
Probability:
20/38
18/38
Mean (Expected Value) Of A Discrete
Random Variable
Find the Mean for Apgar
Scores!
Standard Deviation (and Variance)
for a DRV
Standard Deviation (and
Variance) for a DRV
Find the Standard Deviation for
Apgar Scores!
That was hard!
• Good thing we have a calculator to help reduce
time consumption!
• TI-83
– Start by entering the values of X in L1, and probability
in L2
– 1-var Stats L1, L2
• TI-89
– In the Statistics/List Editor, press F4 (calc) and
choose 1:1-var stats…use the inputs list: list1 and
freq: list 2
Continuous Random Variables
• What if there were infinite probabilities? We cant
add them all up!
• So we look at the area under the curve!
• Why? Well the area under the curve is 1, and
probability adds up to 1, so the area under the
curve can also represent the probability.
• Difference: We cant look at individual
probabilities…we have to look at an interval!
• In fact, all continuous probability models assign
probability 0 to every individual outcome.
Continuous Random Variables
• A continuous random variable X takes 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.
• Lets look at an example of finding the
probability!
Young Women’s Heights
z
x

Objectives
1. Discrete Random Variables
1. What is a discrete random variable?
2. Mean (Expected Value) of a DRV
1.
Examples: Apgar Scores of Babies, Roulette
3. Standard Deviation (and variance) of a DRV
1.
Calculator saves the day!
2. Continuous Random Variables
1. What is a continuous random variable?
1.
Area under the curve!
2. Finding the probability of the interval of outcomes,
Z-scores return!
Homework
Worksheet