Transcript Random Variables

Discrete probability distributions
3)For every possible x value,
0 < P(x) < 1.
4) For all values of x,
S P(x) = 1.
Think About It…

1. In a game of dice a friend gives you a choice. If an
even number is rolled you win $100, if a 5 is rolled you
win $100. Which option would you choose? Why?
1. You friend now states if a 5 is rolled you will win $200.
Which option would you choose? Why?
2. Again, your friend increased a roll of 5 to $300.
Which option would you choose? Why
1. Finally your friend states a roll of 5 will win you $400.
Which option would you choose? Why?
Random Variable A numerical
variable whose
value depends on
the outcome of a
chance experiment
Two types:
 Discrete – count of some
random variable
 Continuous – measure of
some random variable
Random Variable
Example:
Consider tossing a fair coin 3 times.
Define X = the number of heads obtained
DISCRETE
The probabilities pi must satisfy two requirements:
1.Every probability pi is a number between 0 and 1.
• 0 < P(x) < 1.
2.The sum of the probabilities is 1.
• Σ P(x) = 1.
To find the probability of any event, add the probabilities
pi of the particular values xi that make up the event.
Discrete Probability Distribution
 A distribution of a random
variable gives its possible values
and their probabilities.
1) Usually displayed in a table,
but can be displayed with a
histogram or formula
Probability Distribution
Example:
Consider tossing a fair coin 3 times.
Define X = the number of heads obtained
Make a Probability Distribution Chart
Make a Probability Distribution Histogram
What are the chances
 What is P(X > 2)
 Show that this is a legitimate probability
distribution.
Let x be the number of courses for which a
randomly selected student at a certain
university is registered.
X
1 2 3
P(X).02 .03 .09
4 5 6 7
? .40 .16 .05
Why.25
does this not start at zero?
P(x = 4) =
P(x < 4) =
.14
P(x < 4) =
.39
P(x > 5) = .61
What is the probability that the student
is registered for at least five courses?
 Example: Babies’ Health at Birth
Read the example on page 343.
(a) Show that the probability distribution for X is legitimate.
(b) Make a histogram of the probability distribution. Describe what you
see.
(c) Apgar scores of 7 or higher indicate a healthy baby. What is P(X ≥ 7)?
Value:
0
1
2
3
4
5
6
7
8
9
10
Probabilit
y:
0.00
1
0.00
6
0.00
7
0.00
8
0.01
2
0.02
0
0.03
8
0.09
9
0.31
9
0.43
7
0.05
3
(a) All probabilities
are between 0 and
1 and they add up
to 1. This is a
legitimate
probability
distribution.
(c) P(X ≥ 7) = .908
We’d have a 91 %
chance of randomly
choosing a healthy
baby.
(b) The left-skewed shape of the distribution suggests a randomly
selected newborn will have an Apgar score at the high end of
the scale. There is a small chance of getting a baby with a score
of 5 or lower.
Formulas for mean & variance

  x
x 
x
2
xi p
i
i
 x  p
2
i
Found on formula card!
DICE
Expected Value Comparisons
Tebow Time!
The NFL Draft is an annual event which is the most common
source of player recruitment. In the first round of the 2010 NFL draft
the Denver Broncos selected Tim Tebow. At the position of
Quarterback Tebow’s ability was highly debated on a national level.
The Broncos’ Franchise took a major risk, however, do no think for a
second this was not a calculated risk.
Imagine you are on the Broncos Management.
Judging by his record in College, analysts
predict Tebow has a 10% chance of
becoming an elite quarterback, pulling in
$20 million for the franchise. He has a 40%
chance of being average, bringing in $10
million. Otherwise, he will be 2nd or 3rd string
which brings in no money and would be a
loss (the cost of the contract) of $9.7
million.
 Example: Apgar Scores – What’s Typical?
Consider the random variable X = Apgar Score
Compute the mean of the random variable X and interpret it in context.
Value:
0
1
2
3
4
5
6
7
8
9
10
Probability:
0.001
0.006
0.007
0.008
0.012
0.020
0.038
0.099
0.319
0.437
0.053
mx = E(X) = å x i pi
= (0)(0.001) + (1)(0.006) + (2)(0.007) + ...+ (10)(0.053)
= 8.128
The mean Apgar score of a randomly selected newborn is 8.128. This is the longterm average Agar score of many, many randomly chosen babies.
Note: The expected value does not need to be a possible value of X or an integer!
It is a long-term average over many repetitions.


What is the probability that a student has a grade point of 3
or better in this class?

Draw a probability histogram to picture the probability
distribution of the random variable X.
2. Put all the letters of the alphabet in a hat. If you
choose a consonant, I pay you $1. If you choose a
vowel, I pay you $5. X is the random variable
representing the outcome of the experiment.


Create the distribution of X
What is your expected payoff (value) in this game?
You Try…

1. A college instructor teaching a large class traditionally
gives 10% A’s, 20% B’s, 45% C’s, 15% D’s, and 10% F’s. If a
student is chosen at random from the class, the student’s
grade on a 4-point scale (A = 4) is a random variable X.
Create the probability distribution of X.

2. Put all the letters of the alphabet in a hat. If you
choose a consonant, I pay you $1. If you choose a
vowel, I pay you $5. X is the random variable
representing the outcome of the experiment.

What is the game’s variance? The Standard deviation?
s X2 = å(x i -m X )2 pi
Let x be the number of courses for which a
randomly selected student at a certain
university is registered.
X
1
2
3
4
5
6
7
P(X) .02 .03 .09 .25 .40 .16 .05
What is the expected value and
standard deviations of this
distribution?
m = 4.66
& s = 1.2018
Is the formula the only way?!?!?!?!!?
STAT, 1:EDIT
L1 = RANDOM VARIABLE ( X )
L2 = PROBABILITY (PI)
STAT, CALC, 1: 1-VAR STATS
2ND STAT L1
2ND STAT L2
1-VAR STATS L1, L2
Let x be the number of courses for which a
randomly selected student at a certain
university is registered.
X
1
2
3
4
5
6
7
P(X) .02 .03 .09 .25 .40 .16 .05
What is the expected value and
standard deviations of this
distribution?
m = 4.66
& s = 1.2018
Box of 20 DVDs, 4 are defective. Select two from the
box without replacement
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
.
random variable?
• What is the variance? The Standard Deviation?
CARS IN A TOWN
X = number of vehicles owned by a household in a random
town
P(0) =.05, P(1) = .45, P(2) = .275, P(3) = .1, P(4) = .075,
P(5) = .05
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
random variable?
• What is the variance? The Standard Deviation?
BOOK EDITOR
X = # of errors that appear on a randomly selected
page of a book
X = 0, 1, 2, 3 ,4
P(0) =.73, P(1) = .16, P(2) = .06, P(3) = .04, P(4) = .01
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
random variable?
• What is the variance? The Standard Deviation?
FLIGHTS FROM LA TO CHICAGO
X = # of flights that are on time out of 3 independent
flights
P(0) =.064, P(1) = .288, P(2) = .432, P(3) = .216
• Identify your random variables.
• Create a Probability Distribution
• What is the mean (expected value) of the discrete
random variable?
• What is the variance? The Standard Deviation?
Just add orLinear
subtractcombinations
the means!
If y  x1  x 2  ...  xn then
 y   x 1   x 2  ...   xn
y 
2
x 1
2
 x 2
 ...
2
  xn
If independent, always
add the variances!
A nationwide standardized exam consists of a
multiple choice section and a free response
section. For each section, the mean and
standard deviation are reported to be
mean
SD
MC
38
6
FR
30
7
If the test score is computed by adding the
multiple choice and free response, then what
is the mean and standard deviation of the
test?
m = 68
& s = 9.2195
Linear function of a random
variable
The mean is
If x is a
random variable
and a and b
changed
by
are numerical
then the
additionconstants,
&
random
variable
y
is
defined
by
multiplication!
The standard deviation
y a 
is bx
ONLY changed by
multiplication!
 and
 y   a bx  a  b x
2
y

2
a bx

2 2
b x
or  y  b  x
Let x be the number of gallons
required to fill a propane tank.
Suppose that the mean and
standard deviation is 318 gal. and
42 gal., respectively. The
company is considering the pricing
model of a service charge of $50
plus $1.80 per gallon. Let y be the
random variable of the amount
billed. What is the mean and
standard deviation for the amount
billed?m = $622.40 & s = $75.60