Chapter 5

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Discrete
Distributions
Chapter 5
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1
Random Variables
 Random variable
a variable (typically represented by x) that
takes a numerical value by chance.
 For each outcome of a procedure, x takes a
certain value, but for different outcomes that
value may be different.
pg. 205
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2
Examples:
 Number of boys in a randomly selected
family with three children.
Possible values: x=0,1,2,3
 The weight of a randomly selected person
from a population.
Possible values: positive numbers, x>0
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3
Examples:
 Genetics: If two pea plants have both green
and yellow pod genes, the probability that
offspring pea plants has a green pod is .75
That is
P(green) = .75
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4
Discrete and Continuous Random
Variables
 Discrete random variable
Either a finite number of values or countable
number of values (resulting from a counting
process)
 Continuous random variable
Infinitely many values, and those values can
be associated with measurements on a
continuous scale without gaps or interruptions
pg. 206
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5
Probability Distributions
 Probability distribution
A description that gives the probability for
each value of the random variable; often
expressed in the format of a table, graph, or
formula
Pea pod histogram on page 207
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6
Tables
Values:
Probabilities:
x
P(x)
0
1/8
1
3/8
2
3/8
3
1/8
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7
Graphs
The probability histogram is very similar to a
relative frequency histogram, but the vertical
scale shows probabilities.
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8
Requirements for
Probability Distribution
∑ P(x) = 1
where x assumes all possible values.
0  P(x)  1
for every individual value of x.
pg. 207
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9
Mean, Variance and
Standard Deviation of a
Probability Distribution
µ = ∑ [x • P(x)]
Mean
σ = ∑ [(x – µ) • P(x)]
Variance
σ = ∑ [x • P(x)] – µ
Variance (shortcut)
2
2
2
σ=
2
2
∑[x 2 • P(x)] – µ 2
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Standard Deviation
10
Using TI-83/84 calculator
• Press the STAT button and choose EDIT
• Enter the x-values into the list L1 and the P(x)
values into the list L2
• Press the STAT button and choose CALC
• Choose
1-Var Stats and press ENTER
• Type in L1 then , (comma) then L2 on that line,
you will see
1-Var Stats L1,L2
• Press ENTER
• You will see x-bar=…, it is actually m (mean)
and sx=…, it is actually s (st. deviation)
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11
Finding the Mean, Variance, and
Standard Deviation
Example 5, page 209
Table 5-1 describes the probability distribution
for the number of peas with green pods among
five offspring peas obtained from parents both
having the green/yellow pair of genes.
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12
Using Excel - Mean
•
•
•
•
Put the x-values in column 1
Put the P(x)-values in column 2
In column 3 enter “=A1*B1”
Copy and paste that cell to the entire column
of data
• At the bottom of column 3 enter
“=sum(c1:cN)” - N is the last row of the data
Table 5-3, pg 209
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13
Using Excel - Mean
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14
Using Excel – Standard Deviation
Continuing
• In column 4 enter “=power(A1-$c$M,2)*B1”
• Copy and paste that cell to the entire column
of data
• At the bottom of column 4 enter
“=sum(d1:dN)” - N is the last row of the data
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15
Using Excel – Standard Deviation
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16
Using Excel to find the mean and
standard deviation of a discrete
probability distribution
More detailed example
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17
Using Excel to find the mean and
standard deviation of a discrete
probability distribution
(1)
Enter the values and their
probabilities as separate columns
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18
Using Excel (continued)
(2) Check to make sure the values of P(x)
add up to 1:
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Using Excel (continued)
(3) In column C, for each value of x,
calculate x*P(x):
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20
Using Excel (continued)
(4) The sum of column C will give the value
of the mean, μ
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21
Using Excel (continued)
(5) In column D, calculate the values of
X2*P(x)
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Using Excel (continued)
(6) Calculate the sum of the squares times
probability - X2*P(x)
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23
Using Excel (continued)
(7) Use the shortcut formula for the
variance:
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24
Using Excel (continued)
(8) Finally, the Standard deviation is just the
square root of the variance:
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25
How to Choose Lottery Numbers
• Note the sidebar on page 209 about choosing
numbers in a lottery.
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26
Roundoff Rule for
2
µ, σ, and σ
Round results by carrying one more decimal
place than the number of decimal places used
for the random variable x.
If the values of x are integers, round µ, σ, and
σ2 to one decimal place.
Pg 210
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27
Identifying Unusual Results
Range Rule of Thumb
According to the range rule of thumb, most
values should lie within 2 standard deviations
of the mean.
We can therefore identify “unusual” values by
determining if they lie outside these limits:
Maximum usual value = μ + 2σ
Minimum usual value = μ – 2σ
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28
Identifying Unusual Results
By Probabilities
Using Probabilities to Determine When Results
Are Unusual:
 Unusually high: a particular value x is
unusually high if P(x or more) ≤ 0.05.
 Unusually low: a particular value x is
unusually low if P(x or fewer) ≤ 0.05.
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29
Uniform Probability Distribution
A uniform probability distribution has the following
property:
P(x) = c
The value c is a constant, so every event is just a likely
as every other event. If there are n events
P(x) = 1
n
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30
Binomial Probability Distribution
A binomial probability distribution results from a
procedure that meets all the following requirements:
1. The procedure has a fixed number of trials.
2. The trials must be independent. (The outcome of
any individual trial doesn’t affect the probabilities
in the other trials.)
3. Each trial must have all outcomes classified into two
categories (commonly referred to as success and
failure).
4. The probability of a success remains the same in all
trials.
pp 218-219
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31
Notation for Binomial
Probability Distributions
S and F (success and failure) denote the two possible
categories of all outcomes; p and q denote the
probabilities of S and F, respectively:
P(S) = p
(p = probability of success)
P(F) = 1 – p = q
(q = probability of failure)
Note that “success” may not be desirable.
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32
Notation (continued)
n
denotes the fixed number of trials.
x
denotes a specific number of successes in n
trials, so x can be any whole number between 0
and n, inclusive.
p
denotes the probability of success in one of the
n trials.
q
denotes the probability of failure in one of the
n trials.
P(x)
denotes the probability of getting exactly x
successes among the n trials.
pg 219
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33
Methods for Finding
Probabilities
We will now discuss three methods for
finding the probabilities corresponding to the
random variable x in a binomial distribution.
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34
Method 1: Using the Binomial
Probability Formula
P(x) =
n-x
x
n!
•p •q
(n – x )!x!
for x = 0, 1, 2, . . ., n
where
n = number of trials
x = number of successes among n trials
p = probability of success in any one trial
q = probability of failure in any one trial (q = 1 – p)
pg 221
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35
Rationale for the Binomial
Probability Formula
P(x) =
n!
n-x
x
•p •q
(n – x )!x!
The number of
outcomes with
exactly x
successes among
n trials
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36
Binomial Probability Formula
P(x) =
n!
•
(n – x )!x!
Number of
outcomes with
exactly x
successes among
n trials
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px •
n-x
q
The probability of
x successes among
n trials for any one
particular order
37
Method 2: Using TI-83/84
• Press 2nd VARS to get the DISTR menu
• Scroll down to binomialpdf( and press
ENTER
• Type in three values: n, p, x (separated by
commas) and close the parenthesis
• You see a line like binomialpdf(10,.3,6)
• Press ENTER and read the probability of the
value x (successes) in n trials
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38
Alternative use of TI-83/84
• Press 2nd VARS to get the DISTR menu
• Scroll down to binomialcdf( and press
ENTER
• Type in three values: n, p, x (separated by
commas) and close the parenthesis
• You see a line like binomialcdf(10,.3,6)
• Press ENTER and read the combined
probability of all values from 0 to x
(i.e., probability that there are at most x
successes)
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39
Method 3: Excel
• Use an Excel canned function
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40
Using Excel (2007 or later) for
Binomial Distributions
(1) With the cell you want highlighted, From
the home tab, click the arrow beside the ∑
button, and select “more functions”
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41
Using Excel (2007 or later) for
Binomial Distributions
(2) In the pop-up window, scroll down and
select the category “statistical”
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42
Using Excel (2007 or later) for
Binomial Distributions
(3) Under “select a function” scroll down to
BINOMDIST and click “ok”
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43
Using Excel (2007 or later) for
Binomial Distributions
(4)
Fill in the values required
Number_s
# of successes
Trials
# of trials
Probability_s
probability of
success
Cumulative
TRUE if you want
the probability of x or fewer
successes, FALSE if you just
want the probability of
exactly x successes.
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44
Using Excel (2007 or later) for
Binomial Distributions
(5) Now look back at the cell you had
highlighted:
From now on, you
can just enter this
short form if you
find it quicker.
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45
Example
An “unfair coin” has a 0.55 probability of
getting heads and is tossed 10 times
• What is the probability of getting exactly 5
heads?
• What is the probability of getting
at least 4 heads?
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46
Example
Probability of heads: p = 0.55
Number of tosses: n = 10
“Exactly 5 heads” → P(5)
P(5) = 0.234
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47
Example
Probability of heads: p = 0.55
Number of tosses: n = 10
“at most 4 heads” → ∑P(4)
∑P(4) = 0.262
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48
Example
An “unfair coin” has a 0.55 probability of
getting heads and is tossed 10 times
p = 0.55
n = 10
• What is the probability of getting exactly 5
heads?
P(5) = 0.234
• What is the probability of getting
at least 4 heads?
∑P(4) = 0.262
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49
Binomial Distribution: Formulas
Mean
µ = n•p
Variance σ2 = n • p • q
Std. Dev. σ =
n•p•q
Where
n = number of fixed trials
p = probability of success in one of the n trials
q = probability of failure in one of the n trials
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50
Interpretation of Results
It is especially important to interpret results. The
range rule of thumb suggests that values are unusual
if they lie outside of these limits:
Maximum usual values = µ + 2
Minimum usual values = µ – 2
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