ECON 240A

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ECON 240A

• Power 5

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Last Week

• Probability
• Discrete Binomial Probability Distribution

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The Normal Distribution
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Outline

• The normal distribution as an approximation to
  the binomial
• The standardized normal variable, z
• sample means

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Probability Density Function
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Cumulative Distribution Function
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Cumulative Distribution Function

• The probability of getting two or less heads in
  five flips is 0.5
• can use the cumulative distribution function
• can use the probability density function and add
  the probabilities for 0, 1, and 2 heads
• the probability of getting two heads or three
  heads is
• can add the probabilities for 2 heads and three
  heads from the probability density function

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Cumulative Distribution Function
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Cumulative Distribution Function

• the probability of getting two heads or three
  heads is
• can add the probabilities for 2 heads and three
  heads from the probability density function
• can use the probability of getting up to 3 heads,
  P(3 or less heads) from the cumulative
  distribution function (CDF) and subtract the
  probability of getting up to one head P(1 or less
  heads

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Probability Density Function
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Cumulative Distribution Function
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For the Binomial Distribution

• Can use a computer as we did in Lab Two
• Can use Tables for the cumulative distribution
  function of the binomial such as Table 1 in the
  text in Appendix B, p. B-1
• need a table for each p and n.

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Normal Approximation to the binomial

• Fortunately, for large samples, we can
  approximate the binomial with the normal
  distribution, as we saw in Lab Two

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Binomial Probability Density Function
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Binomial Cumulative Distribution Function
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The Normal Distribution

• What would the normal density function look like
  if it had the same expected value and the same
  variance as this binomial distribution
• from Power 4, E(h) np 401/220
• from Power 4, VARh np(1-p) 401/21/2 10

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Normal Approximation to the Binomial De Moivre
This is the probability that the number of heads
will fall in the interval a through b, as
determined by the normal cumulative distribution
function, using a mean of np, and a standard
deviation equal to the square root of np(1-p),
i.e. the square root of the variance of the
binomial distribution. The parameter 1/2 is a
continuity correction since we are approximating
a discrete function with a continuous one, and
was the motivation of using mean 19.5 instead 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 normal approximation

• npgt5
• n(1-p)gt5

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The Standardized Normal Variate

• ZN(0, 10
• Ez 0
• VARZ 1

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Normal Variate x

• E(z) 0
• VAR(z) 1

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b
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b
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For the Normal Distribution

• Can use a computer as we did in Lab Two
• Can use Tables for the cumulative distribution
  function of the normal such as Table 3 in the
  text in Appendix B, p. B-8
• need only one table for the standardized normal
  variate Z.

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Sample Means
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Sample Mean Example

• Rate of return on UC Stock Index Fund
• return equals capital gains or losses plus
  dividends
• monthly rate of return equals price this month
  minus price last month, plus dividends, all
  divided by the price last month
• r(t) p(t) -p(t-1) d(t)/p(t-1)

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Rate of Return UC Stock Index Fund,
http//atyourservice.ucop.edu/
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Table Cont.
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Data Considerations

• Time series data for monthly rate of return
• since we are using the fractional change in price
  (ignoring dividends) times 100 to convert to ,
  the use of changes approximately makes the
  observations independent of one another
• in contrast, if we used price instead of price
  changes, the observations would be correlated,
  not independent

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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

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1.74
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What are the properties of this sample mean?
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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 VARcx Ecx - Ecx2
Ecx-Ex2 Ec2x -Ex2 c2 Ex-Ex2 c2
VARx
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Properties of

• Expected value
• Variance

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Central Limit Theorem

• As the sample size grows, no matter what the
  distribution, f, of the rate of return, r, the
  distribution of the sample mean approaches
  normality

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• An interval for the sample mean

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The rate of return, ri , could be distributed as
uniform
fr(i)
r(i)
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And yet for a large sample, the sample mean will
be distributed as normal
b
a
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Bottom Line

• We can use the normal distribution to calculate
  probability statements about sample means

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• An interval for the sample mean

calculate
?, Assume we know, or use sample standard
deviation, s
infer
choose
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Sample Standard Deviation

• If we use the sample standard deviation, s, then
  for small samples, approximately less then 100
  observations, we use Students t distribution
  instead of the normal

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t-distribution
Text p.253 Normal compared to t
t distribution as smple size grows
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Appendix B Table 4 p. B-9