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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
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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4
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
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5
Tables
Values:
Probabilities:
x
P(x)
0
1/8
1
3/8
2
3/8
3
1/8
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6
Graphs
The probability histogram is very similar to a
relative frequency histogram, but the vertical
scale shows probabilities.
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7
Requirements for
Probability Distribution
∑ P(x) = 1
where x assumes all possible values.
0  P(x)  1
for every individual value of x.
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8
Mean, Variance and
Standard Deviation of a
Probability Distribution
µ = σ [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
9
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
σ 2 to one decimal place.
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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 distributino
for the numbe 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*A2”
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)*A2”
• 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 - Mean
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16
How to Choose Lottery Numbers
• Note the sidebar on page 209 about choosing
numbers in a lottery.
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17
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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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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19
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.
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20
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)
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21
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.
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22
Methods for Finding
Probabilities
We will now discuss two methods for
finding the probabilities corresponding
to the random variable x in a binomial
distribution.
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23
Method 1: Using the Binomial
Probability Formula
P(x) =
n!
•
(n – x )!x!
px •
n-x
q
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)
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24
Rationale for the Binomial
Probability Formula
P(x) =
n!
•
(n – x )!x!
px •
n-x
q
The number of
outcomes with
exactly x
successes
among n trials
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25
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
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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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27
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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28
Binomial Distribution: Formulas
Mean
µ = n•p
Variance  2 = n • p • q
Std. Dev.  =
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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29
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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