Discrete Random Variable

1
Discrete Random Variable

• Let X denote the return of the SP 500 tomorrow,
  rounded to the nearest percent
• what are the possibilities, i.e. 0, 1,
• what is the probability of each of the above
  possibilities
• Probability distribution function f(x)
  P(Xx)

2
Probability Distribution
3
Cumulative Distribution Function
4
Discrete Random Variable

• Expectation
• Variance

5
Which Distribution Has Higher Mean?
6
Which Distribution Has Higher Variance?
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Expectation of a Function of a R.V.

• Function g(X)
• What is the expectation E(g(X))?
• General result
• Example call option on the SP 500

8
Binomial Distribution

• Bernoulli distribution
• A r.v. X has two possible outcomes, 0 or 1
• Binomial distribution
• Number of successes that occur in n trials
• Example Ch. 4, 6b

9
Poisson

• A r.v. X takes on values 0, 1, 2, ....
• Poisson distribution if for some l gt 0,
• The Poisson r.v. is an approximation for binomial
  with l np.
• Example how many days in a year will the SP500
  drop more than 1?
• Example 7b

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Geometric

• Independent trials with prob. of success p
• How many trials until a success occurs?
• What happens when n goes to infinity?
• Example how many days until we get a stock
  market drop of 2 or more?

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Negative Binomial

• Independent trials with prob. of success p
• How many trials until r successes occur?
• What happens when n goes to infinity?
• Example how many days until we get three stock
  market drops of 2 or more (not necessarily
  consecutive)?

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Hyper-Geometric

• Choose n balls out of N, without replacement
• m white, N m black
• X number of white balls selected
• Example 8i
• What happens if you choose the n balls with
  replacement?

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Continuous Random Variable

• Let X denote the return of the SP 500 tomorrow,
  no rounding
• what are the possibilities
• what is the probability of each of the above
  possibilities
• Probability density function

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Probability Density Function
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Cumulative Density Function
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Continuous Random Variable

• Expectation
• Variance
• Example Ch5, 1a, 1b, 2a

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Continuous Random Variable

• For any real-valued function g and continuous
  r.v. X
• Example payoff on a call option, 2b

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Which Distribution Has Higher Mean?
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Which Distribution Has Higher Mean?
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Which Distribution Has Higher Variance?
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Skewness
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Kurtosis
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Additional Sample Questions

• Given a discrete probability function (pdf)
  (i.e., all possible outcomes and their
  probabilities), compute the mean and the variance
• Given a graph of several discrete or continuous
  pdf, estimate which ones has the highest mean,
  variance, skewness, kurtosis
• Given two random variables, guess whether they
  have positive or negative covariance and/or
  correlation

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The Uniform Distribution
Example 3b
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The Normal Distribution
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The Normal Distribution
Example Ch 5, 4b, 4e
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Properties of the Normal Distribution
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Normal is an Approximation to Binomial

• Sn number of successes in n independent trials
  with individual prob. of success p.
• The DeMoivre-Laplace limit theorem

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Normal is an Approximation to Binomial
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Lognormal Distribution

• What is the distribution of the SP 500 index
  tomorrow?
• If the return on the SP500 is normally
  distributed, the index itself is lognormally
  distributed

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Lognormal Distribution
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Chi-squared Distribution

• Sum of squared standard normal variables

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F distribution

• Ratio of two independent chi-squared variables
  with degrees of freedom n1 and n2

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t distribution

• Very important for hypothesis testing

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Normal vs. t distribution
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Exponential Distribution

• PDF
• CDF
• Exercise

37
Joint Distributions of R. V.

• Joint probability distribution function
  f(x,y) P(Xx, Yy)
• Example Ch 6, 1c, 1d

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Independence

• Two variables are independent if, for any two
  sets of real numbers A and B,
• Operationally two variables are indepndent iff
  their joint pdf can be separated for any x and
  y

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Joint Distributions of R. V.

• The expectation of a sum equals the sum of the
  expectations
• The variance of a sum is more complicated
• If independent, then the variance of a sum equals
  the sum of the variances

40
Sum of Normally Distributed RV
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Additional Sample Questions

• Find the distribution of a transformation of two
  or more normal random variables
• By looking at a graph of a pdf, guess whether it
  is normal, log-normal, or t-distribution
• What normally distributed random variables do you
  need to construct an F distribution with 3 and 5
  degrees of freedom

42
Conditional Distributions (Discrete)

• For any two events, E and F,
• Conditional pdf
• Examples Ch 6, 4a, 4b

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Conditional Distributions (Discrete)

• Conditional cdf

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Conditional Distributions (Discrete)

• Example what is the probability that the TSX is
  up, conditional on the SP500 being up?

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Conditional Distributions (Continous)

• Conditional pdf
• Conditional cdf
• Example 5b

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Conditional Distributions (Continous)

• Example what is the probability that the TSX is
  up, conditional on the SP500 being up 3?

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Joint PDF of Functions of R.V.

• joint pdf of X1 and X2
• Equations and can be uniquely solved
  for and given by
• and
• The functions and have continuous
  partial derivatives

48
Joint PDF of Functions of R.V.

• Under the conditions on previous slide,
• Insert eq. 7.1, p275
• Example You manage two portfolios of TSX and
  SP500
• Portfolio 1 50 in each
• Portfolio 2 10 TSX, 90 SP 500
• What is the probability that both of those
  portfolios experience a loss tomorrow?

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Joint PDF of Functions of R.V.

• Example 7a uniform and normal cases

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Estimation

• Given limited data we make educated guesses about
  the true parameters
• Estimation of the mean
• Estimation of the variance
• Random sample

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Population vs. Sample

• Population parameter describes the true
  characteristics of the whole population
• Sample parameter describes characteristics of the
  sample
• Statistics is all about using sample parameters
  to make inferences about the population parameters

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Distribution of the Sample Mean

• The sample mean follows a t-distribution

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Confidence Intervals

• We can estimate the mean, but wed like to know
  how accurate our estimate is
• Wed like to put upper and lower bounds on our
  estimate
• We might need to know whether the true mean is
  above certain value, e.g. zero

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Constructing Confidence Intervals

• We already know the distribution of our estimate
  of the mean
• To construct a 95 confidence interval, for
  instance, just find the values that contain 95
  of the distribution

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Constructing Confidence Intervals
falls in this region 95 of the time
2.5 of the distribution
2.5 of the distribution
Critical values
Critical values
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Confidence Intervals and Hypothesis Testing

• The critical values are available from a table or
  in Matlab
• gtgt tinv(.975, n-1)
• If the confidence interval includes zero, then
  the sample mean is not statistically different
  from the population mean we are testing
• One-sided vs. two-sided tests

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Example

• Are the returns on the SP 500 significantly
  above zero?
• Sample mean .23
• Sample standard deviation .59
• Sample size 128
• Compute the test
• At 95 the critical value is 1.98
• Therefore, we reject that the returns are zero

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Distribution of SP500 Returns

• The direct use of historical data requires the
  following assumptions
• The true distribution of returns is constant
  through time and will not change in the future
• Each period represents an independent draw from
  this distribution

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Distribution of Stock Returns
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Distribution of Stock Returns
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Distribution of Stock Returns
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Linear Regression (Harvey 1989)
63
Harvey 1989
GNP Growth
Spread
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Harvey 1989
Regression Line
GNP Growth
Spread
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Regression

• Minimize the squared residuals

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Regression in Matrix Form

• Regression equation
• Minimize the squared residuals