Download Chapter 3: Random Sampling, Probability, and the Binomial

Outline
Chapter 3: Random Sampling, Probability,
and the Binomial Distribution
Part II
I
Density Curves
I
Random Variables
I
The Binomial Distribution
I
Fitting a Binomial Distribution to Data
Eric D. Nordmoe
Math 261
Department of Mathematics and Computer Science
Kalamazoo College
Spring 2009
Random Sampling Model
Random Variables
A random variable is a variable that takes on numerical values
that depend on the outcome of a chance operation.
Probability Model
Types of Random Variables
Random Sampling
Population
Sample
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Continuous random variables take values on a continuous
scale.
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Discrete random variables have a discrete list of possible
values.
Probability Models for Random Variables
Density Curve Example
The method of characterizing the distribution of a random
variable depends on the nature of the variable:
0.005
0.010
The distribution of a continuous random variable is described
by a density curve.
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A probability density may be viewed as an idealized
histogram.
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Continuous Random Variables
Density
The distribution of a discrete random variable is described by
the probability distribution comprised of enumerated possible
values and corresponding probabilities.
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Discrete Random Variables
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A stylistic example
50
100
150
Diastolic Blood Pressure
Histogram
Density Curve
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The proportion of the distribution that falls in that range.
The probability that a randomly selected individual has a
value in that range.
0.25
0.20
The total area underneath the density is 1.
The area under the curve and above any range of values
can be interpreted in two ways:
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0.10
The density curve always lies on or above the horizontal
axis.
P(1<Y<3)= 0.41
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Probability as the Area Under a Density Curve
0.00
Properties of Density Curves
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1
2
3
4
5
6
7
8
9
10
Probability Distribution of Y
yi
Pr(Y = yi )
1
0.1
2
0.2
3
0.3
4
0.4
Total
1.0
Probabilities of all possible values must sum to one.
0.5
Probability
0.2 0.3 0.4
The probability distribution of a discrete random variable is a list
of the possible values of the random variable and the
probability associated with each possible value.
Graphical
0.1
Probability Distributions
Other Views of the Probability Distribution of Y
0.0
Properties of Discrete Random Variables
1
2
3
Formulaic
P(Y = y ) =
Mean and Variance of a Discrete Random Variable
The Mean µY
The mean of a discrete random variable is
X
µY =
yi Pr(Y = yi )
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the sum of the values times their corresponding
probabilities.
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the center of gravity of the probability distribution.
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also known as the expected value E(Y ).
4
Y
y
,
10
y = 1, 2, 3, 4.
Computing the Mean
Using the data from Example 3.35 (p.98) we have:
No. of Vertebrae
y
20
21
22
23
Total
Hence, E(Y ) = µY = 21.49
Pr(y )
0.03
0.51
0.4
0.06
1
y × Pr (y )
0.60
10.71
8.80
1.38
21.49
Mean and Variance of a Discrete Random Variable
Given µY = 21.49, we have:
The Variance σY2
The variance of a discrete random variable is
X
σY2 =
(yi − µY )2 Pr(Y = yi )
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Computing the Variance
the sum of the squared deviations of the values from their
expected value times their corresponding probabilities.
No. of Vert
y
20
21
22
23
Total
Pr(y )
0.03
0.51
0.4
0.06
1
(y − 21.49)2
2.220
0.240
0.260
2.280
Hence, σY2 = 0.430 and σ =
Properties of Means and Variances
(y − 21.49)2 × Pr(y )
0.067
0.122
0.104
0.137
0.430
√
0.430 = .656.
Properties of Means and Variances
Variances
Means
1. The mean of a sum is the sum of the means.
µX +Y = µX + µY
1. The variance of a sum is the sum of the variances if the
random variables are independent
σX2 +Y = σX2 + σY2
2. The mean of a difference is the difference of the means.
µX −Y = µX − µY
3. Means of linear combinations are the same linear
combination of the mean:
2. The variance of a difference is the sum of the variances if
the random variables are independent
σX2 −Y = σX2 + σY2
3. The variance is not changed by adding a constant:
µaX +b = aµX + b
σX2 +b = σX2
4. The variance of a linear combination
2
2 2
σaX
+b = a σX .
Binomial Random Variables
Binomial Random Variables
Binomial Experiments
Random variable
The binomial random variable tallies the number of successes
in the n trials.
1. Binary outcomes: Each trial has just two possible
outcomes (success or failure).
Parameters
2. Independent trials
The binomial distribution has two parameters:
3. n is fixed: The number of trials is fixed in advance
n the number of trials
4. Same value of p: each trial has the same success
probability, p.
p the success probability
Examples
Binomial Probability Distributions
Binomial Probability Distributions
Binomial Distribution
n = 100 , p = 0.5
Binomial Distribution
n = 30 , p = 0.1
Binomial Distribution
n = 100 , p = 0.1
0
2
4
6
Possible Values
8
10
0
5
10
15
20
Possible Values
25
30
0
20
40
60
Possible Values
80
100
0.10
0
2
4
6
Possible Values
8
10
0.08
Probability
0.00
0.00
0.0
0.00
0.00
0.00
0.02
0.05
0.05
0.02
0.1
0.04
0.06
Probability
0.05
0.10
0.10
0.2
Probability
0.04
Probability
Probability
Probability
0.15
0.15
0.10
0.06
0.3
0.20
0.20
0.12
0.08
0.25
Binomial Distribution
n = 10 , p = 0.1
0.4
Binomial Distribution
n = 30 , p = 0.5
0.15
Binomial Distribution
n = 10 , p = 0.5
0
5
10
15
20
Possible Values
25
30
0
20
40
60
Possible Values
80
100
Binomial Probability Distributions
Binomial Distribution
n = 30 , p = 0.9
Binomial Distribution
n = 100 , p = 0.9
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0.4
Binomial Distribution
n = 10 , p = 0.9
The Binomial Probability Distribution Function
0.12
Pr{j successes} = Pr(Y = j) = n Cj pj (1 − p)n−j
0.20
where
n Cj
Probability
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=
n!
j!(n − j)!
The binomial coefficient n Cj is the number of ways to get j
successes in n trials.
0
2
4
6
8
10
0.00
0.0
0.00
0.02
0.05
0.1
0.04
0.10
0.06
Probability
0.08
0.15
0.10
0.3
0.2
Probability
The binomial probability distribution function is
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5
Possible Values
10
15
20
25
30
Possible Values
0
20
40
60
80
100
Possible Values
The Binomial Probability Distribution Function
Mean and Variance
The mean (or expected value) of a binomial random variable is
E(Y ) = µY = np
and the variance is
V (Y ) = σ 2 = np(1 − p)
so the standard deviation is
σY =
p
np(1 − p.
Binomial Data
Number of Boys Data. The following table shows the number of
boys among the first four children in 3343 Swedish families of
size 4 or more.
Number of Boys
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1
2
3
4
Total
Frequency
183
789
1250
875
246
3343