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ECON 240AECON 240A
Power 5Power 5
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Last WeekLast Week
ProbabilityProbability Discrete Binomial Probability Discrete Binomial Probability
DistributionDistribution
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The Normal DistributionThe Normal Distribution
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OutlineOutline
The normal distribution as an The normal distribution as an approximation to the binomialapproximation to the binomial
The standardized normal variable, The standardized normal variable, zz
sample meanssample means
Binomial Distribution
Five Flips of a Fair Coindensity cumulative
k n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 1
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Probability Density Probability Density FunctionFunction
Five Flips of a Fair Coin
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5
Number of Heads
Pro
bab
ilit
y
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Cumulative Distribution Cumulative Distribution FunctionFunction
Five Flips of a Fair Coin
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
Number of Heads
Pro
ba
bil
ity
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Cumulative Distribution Cumulative Distribution FunctionFunction
The probability of getting two or less The probability of getting two or less heads in five flips is 0.5heads in five flips is 0.5• can use the cumulative distribution functioncan use the cumulative distribution function• can use the probability density function and can use the probability density function and
add the probabilities for 0, 1, and 2 headsadd the probabilities for 0, 1, and 2 heads the probability of getting two heads or the probability of getting two heads or
three heads isthree heads is• can add the probabilities for 2 heads and can add the probabilities for 2 heads and
three heads from the probability density three heads from the probability density function function
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Cumulative Distribution Cumulative Distribution FunctionFunction
Five Flips of a Fair Coin
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
Number of Heads
Pro
ba
bil
ity
Binomial Distribution
Five Flips of a Fair Coindensity cumulative
k n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 1
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Cumulative Distribution Cumulative Distribution FunctionFunction
the probability of getting two heads or the probability of getting two heads or three heads is:three heads is:• can add the probabilities for 2 heads and can add the probabilities for 2 heads and
three heads from the probability density three heads from the probability density function function
• can use the probability of getting up to 3 can use the probability of getting up to 3 heads, P(3 or less heads) from the heads, P(3 or less heads) from the cumulative distribution function (CDF) and cumulative distribution function (CDF) and subtract the probability of getting up to subtract the probability of getting up to one head P(1 or less heads]one head P(1 or less heads]
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Probability Density Probability Density FunctionFunction
Five Flips of a Fair Coin
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5
Number of Heads
Pro
bab
ilit
y
Binomial Distribution
Five Flips of a Fair Coindensity cumulative
k n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 1
Binomial Distribution
Five Flips of a Fair Coindensity cumulative
k n p0 5 0.03125 0.031251 5 0.15625 0.18752 5 0.3125 0.53 5 0.3125 0.81254 5 0.15625 0.968755 5 0.03125 1
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Cumulative Distribution Cumulative Distribution FunctionFunction
Five Flips of a Fair Coin
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 1 2 3 4 5
Number of Heads
Pro
ba
bil
ity
]3[ hp
]1[ hp
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For the Binomial For the Binomial DistributionDistribution
Can use a computer as we did in Can use a computer as we did in Lab TwoLab Two
Can use Tables for the cumulative Can use Tables for the cumulative distribution function of the distribution function of the binomial such as Table 1 in the binomial such as Table 1 in the text in Appendix B, p. B-1text in Appendix B, p. B-1• need a table for each p and n.need a table for each p and n.
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Normal Approximation to Normal Approximation to the binomialthe binomial
Fortunately, for large samples, we Fortunately, for large samples, we can approximate the binomial with can approximate the binomial with the normal distribution, as we saw the normal distribution, as we saw in Lab Twoin Lab Two
Forty Tosses of a Fair Coin
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 3 6 9 12 15 18 21 24 27 30 33 36 39
Number of Heads
Pro
bab
ilty
Binomial Probability Density Function
Forty Tosses of a Fair Coin
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 3 6 9 12 15 18 21 24 27 30 33 36 39
Number of Heads
Pro
bab
ility
Binomial Cumulative Distribution Function
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The Normal DistributionThe Normal Distribution
What would the normal density What would the normal density function look like if it had the same function look like if it had the same expected value and the same expected value and the same variance as this binomial variance as this binomial distributiondistribution• from Power 4, E(h) = n*p =40*1/2=20from Power 4, E(h) = n*p =40*1/2=20• from Power 4, VAR[h] = n*p*(1-p) = from Power 4, VAR[h] = n*p*(1-p) =
40*1/2*1/2 =1040*1/2*1/2 =10
Normal Density Function, Mean 20, Variance 10
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50
Number of Heads
Den
sity
Comparing the Normal Density with the Binomial Probability Distribution
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 3 6 9 12 15 18 21 24 27 30 33 36 39
Number of Heads
Den
sity
binomial
normal
Comparing the Binomial and Normal Distribution Functions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 3 6 9 12 15 18 21 24 27 30 33 36 39
Number of Heads
Pro
bab
ilit
y
binomial
normal
Comparing the Binomial and Normal (Mean = 19.5) Cumulative Distribution Functions
0
0.2
0.4
0.6
0.8
1
1.2
0 3 6 9 12 15 18 21 24 27 30 33 36 39
Number of Heads
Pro
bab
ilit
y
binomial
normal
Normal Approximation to Normal Approximation to the Binomial: De Moivrethe Binomial: De Moivre
P(a k b) P[(a – ½ - np)/ )1(pnp z [(b + ½ - np)/ )1(pnp]
This is the probability that the number of heads will fall in theinterval a through b, as determined by the normal cumulative distribution function, using a mean of n*p, and a standard deviation equal to the square root of n*p*(1-p), i.e. the square root of thevariance of the binomial distribution. The parameter 1/2 is a continuity correction since we are approximating a discrete functionwith a continuous one, and was the motivation of using mean 19.5instead of mean 20 in the previous slide. Visually, this seemed to be a better approximation than using a mean of 20.
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Guidelines for using the Guidelines for using the normal approximationnormal approximation
n*p>=5n*p>=5 n*(1-p)>=5n*(1-p)>=5
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The Standardized Normal The Standardized Normal VariateVariate
Z~N(0, 10]Z~N(0, 10] E[z} = 0E[z} = 0 VAR[Z] = 1VAR[Z] = 1
Density Function for the Standardized Normal Variate
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
-5 -4 -3 -2 -1 0 1 2 3 4 5
Standard Deviations
Den
sity
2]1/)0[(2/1*]2/1[)( zezf
Cumulative Distribution Function for a Standardized Normal Variate
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-5 -4 -3 -2 -1 0 1 2 3 4 5
Standrd Deviations
Pro
ba
bil
ty
Normal Variate xNormal Variate x
– E(z) = 0E(z) = 0– VAR(z) = 1VAR(z) = 1
),(~ 2Nx
)(xE
2][ xVAR
xorzxz *,/)(
Normal Density Function, Mean 20, Variance 10
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50
Number of Heads
Den
sity
2]/)[(2/1*)2/1()( xexf
Normal Cumulative Distribution Function, Mean=20, Variance = 10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 10 20 30 40 50
Number of Heads
Pro
ba
bil
ity
b
dxxfbF )()(
b
Normal Density Function, Mean 20, Variance 10
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50
Number of Heads
Den
sity
2]/)[(2/1*)2/1()( xexf
b
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For the Normal For the Normal DistributionDistribution
Can use a computer as we did in Can use a computer as we did in Lab TwoLab Two
Can use Tables for the cumulative Can use Tables for the cumulative distribution function of the normal distribution function of the normal such as Table 3 in the text in such as Table 3 in the text in Appendix B, p. B-8Appendix B, p. B-8• need only one table for the need only one table for the
standardized normal variate Z.standardized normal variate Z.
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Sample MeansSample Means
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Sample Mean ExampleSample Mean Example
Rate of return on UC Stock Index FundRate of return on UC Stock Index Fund return equals capital gains or losses return equals capital gains or losses
plus dividendsplus dividends monthly rate of return equals price monthly rate of return equals price
this month minus price last month, this month minus price last month, plus dividends, all divided by the plus dividends, all divided by the price last monthprice last month
r(t) ={ p(t) -p(t-1) + d(t)}/p(t-1)r(t) ={ p(t) -p(t-1) + d(t)}/p(t-1)
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Rate of Return UC Stock Rate of Return UC Stock Index Fund,Index Fund, http://atyourservice.ucop.http://atyourservice.ucop.edu/edu/
Date RateSept 2002 -3.51Oct 2002 7.84Nov 2002 5.13Dec 2002 -5.24Jan 2003 -2.51Feb 2003 -1.66March 2003 0.61
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Table Cont.Table Cont.
Date Rate
April 2003 8.43
May 2003 5.55
June 2003 1.34
July 2003 2.67
Aug 2003 2.26
Rate of Return UC Stock Index Fund
-6
-4
-2
0
2
4
6
8
10
Jul-02 Sep-02
Oct-02 Dec-02
Jan-03 Mar-03
May-03
Jun-03 Aug-03
Oct-03
Date
Rat
e
42
Data ConsiderationsData Considerations
Time series data for monthly rate of Time series data for monthly rate of returnreturn
since we are using the fractional change since we are using the fractional change in price (ignoring dividends) times 100 in price (ignoring dividends) times 100 to convert to %, the use of changes to convert to %, the use of changes approximately makes the observations approximately makes the observations independent of one anotherindependent of one another
in contrast, if we used price instead of in contrast, if we used price instead of price changes, the observations would price changes, the observations would be correlated, not independentbe correlated, not independent
Cont.
assume a fixed target, i.e. the central tendency of the rate is fixed, not time varying
Assume the rate has some distribution, f, other than normal: ri
= sample mean:
),( 2f
74.112/)(03'
02'
Aug
Septi
irr
Rate of Return UC Stock Index Fund
-6
-4
-2
0
2
4
6
8
10
Jul-02 Sep-02
Oct-02 Dec-02
Jan-03 Mar-03
May-03
Jun-03 Aug-03
Oct-03
Date
Rat
e
1.74
45
What are the properties of What are the properties of this sample mean?this sample mean?
n
i
n
i
ixEcixfixcixfixccxE11
)]([*)]([*)(*)]([*)(*][
Note: Expected value of a constant, c, times a random variable, x(i), where i indexes the observation
Note: Variance of a constant times a random variable
VAR[c*x] = E{cx - E[c*x]}2 = E{c*[x-Ex]}2 =E{c2[x -Ex]2 } = c2 *E[x-Ex]2 = c2 *VARx
Properties of
Expected value:
Variance
12
1
12
1
12*)12/1()]([12/1}12/)]([{)(i i
irEirErE
12
1
12
1
2222 12/12*)12/1()]([)12/1(}12/)({)(i i
irVARirVARrVAR
),(~)(,, 2firwherer
48
Central Limit TheoremCentral Limit Theorem
As the sample size grows, no As the sample size grows, no matter what the distribution, f, of matter what the distribution, f, of the rate of return, r, the the rate of return, r, the distribution of the sample mean distribution of the sample mean approaches normalityapproaches normality
An interval for the sample mean
]}/)[(]/){[()( nbrnapbrap
50
The rate of return, rThe rate of return, ri i , could be distributed as uniform
f[r(i)]
r(i)
51
And yet for a large sample, And yet for a large sample, the sample mean will be the sample mean will be distributed as normaldistributed as normal
Cumulative Distribution Function for a Standardized Normal Variate
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-5 -4 -3 -2 -1 0 1 2 3 4 5
Standrd Deviations
Pro
ba
bilt
y
a
b
52
Bottom LineBottom Line
We can use the normal distribution We can use the normal distribution to calculate probability statements to calculate probability statements about sample meansabout sample means
An interval for the sample mean
]}/)[(]/){[()( nbrnapbrap
calculate
chooseinfer
?, Assume we know,or use sample standard deviation, s
Sample Standard Deviation
If we use the sample standard deviation, s, then for small samples, approximately less then 100 observations, we use Student’s t distribution instead of the normal
n
i
nrirs1
2 )1/(])([
55
Text p.253Normal compared to t
t-distribution
t distributionas smple size grows
56
Appendix BTable 4p. B-9
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