1. Introduction to Probability Theory,
Statistics and Distributions
FRM Level 1 Part 2
Source Material ‐ https://www.garp.org/#!/frm

2. Probability Theory

3. Probability Functions
• Probability function p(x), gives the probability
that a discrete random variable will take on a
value x [eg. p(x) = x / 15 for X = {1,2,3,4,5} ‐> p(3)
= 20%.
• Probability Density Function (PDF) f(x), gives the
probability of a continuous random variable.
• Cumulative distribution function (CDF) F(x), gives
the probability that a random variable will be less
than or equal to a given value.

4. Discrete Uniform Distribution
• Properties
– Finite number of possible outcomes will equal
probability
• Example
– p(x) = .2 for X = {1,2,3,4,5}

5. Probability Terms
• Unconditional probability (aka. marginal
probability): Probability of an event regardless
of past, future, or other events
• Conditional probability P(A|B): Probability of
some event A given (or conditional upon) an
event B
• Joint probability P(AB): Probability of both
events A and B occurring. P(AB)=P(A|B) x P(B)

6. Probability Terms
• At least one occurrence P(A or B):
– P(A or B) = P(A) + P(B) – P(AB)
Don’t double
count P(AB)

7. Basic Statistics

8. Geometric vs Arithmetic Mean
• Geometric mean: used to calculate periodic
compound growth rates
• Arithmetic mean (i.e. simple average) will
equal geometric if sample has no variability
• The greater the variability in the sample the
more arithmetic mean will exceed geometric.
• Geometric mean formula with sample of n
returns R

9. Expected Value
• Expected Value E() is the average (i.e. mean)
• Properties:
1. If c is any constant: E(cX) = cE(X)
2. If X and Y are any random variables: E(X+Y) = E(X)
+ E(Y)
3. If c and a are constants: E(cX+a) = cE(X)+a
4. If X and Y are independent random variables:
E(XY) = E(X) x E(Y)

10. Variance and Standard Deviation
• Variance = σ2 = E[(X – E(X))2], properties:
1. If c is constant: Var(c) = 0 and VaR(cX)=c2 x Var(X)
2. If c and a are constants: Var(aX+c ) = a2 x Var(X)
3. If c and a are constants: E(cX+a) = cE(X)+a
4. If X and Y are independent random variables:
Var(X+Y) = Var(X‐Y) = Var(X) + Var(Y)
• Standard Deviation = Square root of Variance
= σ = {E[(X – E(X))2]}1/2

11. Sample Variance & Standard Deviation
• Key difference between calculating sample
variance s2 and standard deviation s and
population variance σ2 and standard deviation
σ is that the sum of squared deviations for
sample statistics is divided by n‐1 instead of n

12. Covariance
• Covariance: A measure of how to variables move together.
Cov(X,Y) = E[(X‐E(X))(Y‐E(Y))] = E(XY)‐E(X)E(Y)
• Interpretation:
– Values range from negative to positive infinity.
– Positive (negative) covariance means when one variable has
been above its mean the other variable has been above (below)
its mean.
– Units of covariance are difficult to interpret which is why we
more commonly use correlation (next slide)
• Properties:
– If X and Y are independent then Cov(X,Y) = 0
– Covariance of a variable X with itself is Var(X)
– If X and Y are NOT independent:
• Var(X+Y) = Var(X) + Var(Y) + 2(Cov(X,Y)
• Var(X‐Y) = Var(X) + Var(Y) ‐ 2(Cov(X,Y)

13. Correlation
• Correlation: A standardized measure of the
linear relationship between two variables
Corr(X,Y)
,
• Properties
– Correlation has no units
– Values range from 1 (perfect positive correlation)
to ‐1 (perfect negative correlation)

14. Correlation Examples

15. Variance: 2nd Central Moment Mean: 1st Raw Moment
Moments
• Moments are descriptions of probability distributions and PDFs:
• Central Moments (as opposed to Raw Moment) are measures
relative to the mean. The kth central moment is::
Skewness: Standardized 3rd Central Moment Kurtosis: Standardized 4th Central Moment

16. Skewness
• Positive (negative) skew has outliers on the
right (left) tail
• Skew absolute values > 0.5 are generally
considered significant

17. Kurtosis
• Measures the degree to which a distribution is
more peaked with fatter tails (leptokurtic) or less
peaked with thinner tails (platykurtic) than a
normal distribution.
• Kurtosis of normal
distribution is 3.0
• Excess kurtosis = kurtosis
minus 3 (so normal
distribution has zero excess
kurtosis
• Excess kurtosis > 1 or < ‐1 is
considered significant

18. Co‐skewness and Co‐kurtosis
• Similar to moments and central moments for
means and variance, we can identify cross
central moments for the concept of
covariance:
– Coskewness: 3rd cross central moment
– Cokurtosis: 4th cross central moment
• Coskewness and cokurtosis can be captured
by incorporating time‐varying volatility and/or
time‐varying correlation into risk models

19. Desirable Estimator Properties
• Unbiased: Expected value equal to parameter
• Efficient: Sampling distribution has smallest
variance of all unbiased estimators
• Consistent: Larger sample ‐> better estimator.
Standard error of estimate decreases with
larger sample size
• Linearity: Used as a linear function of sample
data

20. Distributions

21. Uniform Distribution
• Continuous Uniform Distribution (CUD) is
defined over a range that spans between
some lower limit, a, and some upper limit, b,
which are the only two parameters.
• Properties
– For a random variable x following a CUD, a<x<b
– P(X=x)=0 because it is continuous

22. Binomial Distribution
• A random variable following a BD is defined as
the number of “successes” (x) in a given number
of trials (n) whereby the outcome can be either a
success or a failure:
!
! !
• Mean = np
• Variance = np(1‐p)
Number of ways to
choose x from n
P = Probability
of success on
each trial
(1‐P) = Probability of
failure on each trial

23. Poisson Distribution
• A discrete frequency distribution that gives the
probability of a number of independent events, x,
occurring in a fixed time. The only parameter is λ
which refers to the expected number of events in
the same fixed time.
• An example of a Poisson Distribution is the
number of fraudulent loans in an acquired
portfolio.
• Mean and Variance are equal to the parameter, λ

24. Binomial vs Poisson
• If an exact probability of an event happening is
given, or implied, in the question, and you are
asked to calculate the probability of this event
happening k times out of n, then the Binomial
Distribution must be used.
• If a mean or average probability of an event
happening per unit time/per page/per mile
cycled etc., is given, and you are asked to
calculate a probability of n events happening in a
given time/number of pages/number of miles
cycled, then the Poisson Distribution is used.

25. Standard Normal Distribution
• A standard normal distribution is a normal distribution that has
been standardized so that mean = 0 and standard deviation = 1
• A normal distribution is completely described by mean and variance
• Skewness = 0, Kurtosis = 3
• Linear combination of normally distributed random variables is also
normally distributed
• Multivariate Normal: more than one random variable, correlation
between outcomes
E(x)

26. Confidence Interval
• Confidence Interval: A range of values around an expected outcome within we
expect the actual outcome to occur some specified percent of the time. Example
below for a normal distribution
• Common Standard Normal Z‐Scores
Two‐sided
– N(‐1.96) =2.5% so 5% in two‐tails
– N(‐2.58) = 0.005, so 1.0% in two‐tails
One‐sided
– N(‐1.645) = 5%
– N(‐2.33) = 1%.
.

27. Calculating probabilities
• Example: The EPS for a large group of firms are normally distributed and has mean
= $4 and standard deviation of $1.50. Find the probability that a randomly selected
firm’s earnings are less than $3.70
• Z = (3.7 – 4) / 1.5 = ‐2
• 3.7 is .2 standard deviations below mean of 4
• Excerpt from Table of Cumulative Probabilities for a Standard Normal Distribution
provides probability for area to right of ‐0.2 standard deviations
• Take 1 – 0.5793 to get probability of earning’s less than $3.70

28. Lognormal Distribution
• The lognormal distribution is generated by the function
ex where x is normally distributed
• Distribution is skewed right and bounded to the left by
zero
Example: Stock Returns Stock Price

29. Student’s t‐Distribution
• Symmetrical and bell shaped
• Less peaked and fatter tails than normal distribution
• Defined by single parameter, degrees of freedom (df)
where df = n‐1
• As df increases, t‐distribution approached normal
distribution

30. Chi‐Squared Distribution
• Asymetrical distribution bounded by zero and
approaches normal distribution as degrees of freedom,
k, increase.
• Often used for hypothesis testing of the population
variance (which is always positive or zero)

31. F‐Distribution
• Right skewed distribution bounded by zero and approaches normal
distribution as degrees of freedom, k, increase.
• Shape is determined by two separate degrees of freedom.
• Often used to hypothesis testing of the equality of the variances of
two populations. In this case, the two separate degrees of freedom
are taken from the sample variance compared in the test.

32. Bayesian Analysis

33. Bayes’ Theorem
• Bayes’ Theorem is used to update a given set
of prior probabilities for a given event in
response to the arrival of new information

34. Probability Matrix
• A set of conditional probabilities (inside
matrix) and unconditional probabilities
(outside matrix)

35. Bayes’ vs Frequency
• Bayesian Approach: Assume priors and update
using new information
• Frequentist Approach: Do not impose priors
on the data. Use the probabilities implied by
what information is available.
• Rule of thumb:
– Use Bayesian approach when lacking data
– Use Frequentist approach with large data

36. Central Limit Theorem and Sample
Means

37. Central Limit Theorem
• For any population with a well defined mean,
µ, and variance, σ2, as the size of the random
sample, n, gets large the distribution of
sample means approaches a normal
distribution with the same mean µ and
variance σ2 / n.
• Allows us to infer about population means
using sample means.

38. Standard Error of the Sample Mean
• Standard Error of the sample mean is the standard
deviation of the distribution of sample means.
– When population σ is known:
– When population σ is unknown:
• Example: The mean P/E for a sample of 41 firms is 19
and the standard deviation of the population is 6.6.
What is the standard error of the sample mean?

39. Confidence Intervals for Sample Mean
• If the population has a normal distribution with a known variance, σ, the
confidence interval for the population mean can be established as follows:
– Sample Mean +/‐ /
σ
• If the population has a normal distribution and only the sample variance, s, is
known the confidence interval for the population mean should be constructed
using a t‐distribution
– Sample Mean +/‐ /
s

40. Hypothesis Testing and Confidence
Intervals

41. 1 vs 2 tailed test
• Two tailed test
• Use when testing to see if a
population parameter is
different from a specified value
• H0: μ = 0 vs HA: μ ≠ 0
• One tailed test
• Use when testing to see if a
population parameter is above
or below a specified value, Ex:
• H0: μ ≤ 0 vs HA: μ > 0
• Type I Error: rejection of the null hypothesis when it is actually true
• Type II Error: failure to reject the null hypothesis when it is actually false

42. Test Statistic and P‐Value
• Test Statistic: Calculated from sample data and compared to critical
value(s) to test H0
• P‐Value: Probability of obtaining critical value that is the same as the
computed test statistic. It is the smallest level of significance for which the
null hypothesis can be rejected.
• Example: Test if population bank mean deposit decay rate is > 1%
– H0: μ ≤ 0.01 vs HA: μ > 0.01 || Type of test: One tailed
– Facts: n (banks) = 25, µ (sample mean) = 1.5%, s (sample standard deviation) = 1.4%
• Steps:
1. Select test statistic (t‐stat)
2. Specify significance level (5%)
3. Determine Critical Value
4. Calculate Test Statistic (below)
5. Decision: Reject H0: μ ≤ 0.01
= 1.711 = 1.785

43. • Is the variance of a banks trading book returns = 0.16%?
– H0: σ2 = 0.16% vs HA: σ2 ≠ 0.16% || Type of test: Two tailed
– Facts: n (months) = 24, s2 (sample average standard deviation) = 0.1444%
• Steps:
1. Select test statistic (Chi‐square)
2. Specify significance level (5%)
3. Determine Critical Values
4. Calculate Test Statistic (below)
5. Fail to Reject H0: μ ≤ 0.016
= (23 x .14%) / .16% = 20.75
Chi‐Square test of population variance
df 0.975 0.025
22 10.982 36.781
23 11.689 38.076
24 12.401 39.364
Chi Square Table
Chi Square PDF Distribution
11.689 ‐ Critical Values – 38.076
X2 = 20.75
Intuition: X2 is close to n
(24) because hypothesized
σ2 is close to observed s2

44. • An F‐test is any statistical test in which the test statistic has an F‐distribution under the null
hypothesis.
• Often used to determine the best of two statistical models by identifying the one that best
fits the data they were both estimated upon.
• Tests whether any independent variables explain variation in dependent variable.
F‐statistic with k and n – (k+1) degrees of freedom
• k = number of independent variables (attributable to ESS)
• n – (k+1) = observations minus number of coefficients
Example:
• The ESS and SSR from a model are 500 and 200 respectivly
• Sample observations = 100, Model has 3 variables
• F = (500 / 3) / [200 /(100‐3‐1)] = 80
• Critical 95% F‐Value for 3 and 96 df = 2.72
F‐Test
Numerator Degrees
of Freedom
Denominator Degrees
of Freedom

45. Chebyshev’s inequality
• States that for any set of observations, whether sample or
population data and regardless of the shape of the
distribution, the percentage of the observations that lie
within k standard deviations of the mean is at least:
1 – 1/k2 for all k > 1
• Example: What is the minimum percentage of any
distribution that will lie within +/‐ 2 standard deviations of
the mean?
1 – 1/(2x2) = 75%

46. Copulas

47. Bivariate Normal Sample
• Steps for generating two correlated variables, each with a standard normal
distribution (SND):
1. Draw independent samples of two SND, ZX & ZY.
2. This creates normally distributed error terms EX and Ey .Keep the error term
for variable X.
3. Change the error term for variable Y using the following formula:
Desired
correlation
between X and Y
ZX ZY EX EY
1 1.20 0.04 1.20 0.64
2 0.35 ‐0.64 0.35 ‐0.38
3 0.76 ‐1.50 0.76 ‐0.91
997 ‐0.99 ‐1.03 ‐0.99 ‐1.39
998 ‐1.15 ‐0.46 ‐1.15 ‐0.97
999 ‐2.17 2.30 ‐2.17 0.91
1000 ‐0.14 ‐0.22 ‐0.14 ‐0.26
0.5 Correlation

48. Factor Models
• Factor models can be used to define correlations between normally distributed
variables.
• Equation below is for a one‐factor model where each Ui has a component
dependent on one common factor F in addition to another idiosyncratic factor Zi
that is uncorrelated with other variables.
• Steps to construct:
1. Create the SND common factor F
2. Choose a weight α for each Ui
3. Create correlations with F (previous slide)
4. Draw i number of SND idiosyncratic factors Z
5. Calculate U using equation to right
• Advantages of Single Factor Models:
– Covariance matrix is positive‐semidefinite
– Number of correlation estimations is reduced to N from [Nx(N‐1)]/2
• Capital Asset Pricing Model (CAPM) is well known example of Single Factor Model
Common
Factor
Idiosyncratic
Factor

49. Copulas
• Copula functions are joint probability functions between
multiple variables that allow the individual variable
behavior (e.g. marginal distributions) to remain intact
• Key property of copula functions is that they allow the
introduction of correlation while preserving marginal
distributions
• Gaussian copula maps the marginal distribution of each
variable to a bivariate standard normal distribution
• Student’s t‐copula is similar to the Gaussian copula except
that the variables are mapped to a bivariate Student t‐
distribution. As with the marginal Student t‐distribution,
the tails are fatter than with a normal distribution which
makes it more conservative choice because it increases the
implied probability of joint extreme events

50. • Both graphs show different but stable correlations across the joint distributions. However, this
Gaussian Copula assumption that correlation in the tail region is the same as the correlation
throughout the entire joint distribution may not be correct.
50
Standard Bivariate Normal : Correlation = 0.75Standard Bivariate Normal : Correlation = 0.25
x1
x2
x1
x2
Flaws with Gaussian Copula
Standard Bivariate Normal: Correlation = 0.25
Weak Correlation: Lower chance that 2 bad
outcomes occur simultaneously.
Standard Bivariate Normal: Correlation = 0.75
Strong Correlation: Higher chance that 2 bad
outcomes occur simultaneously

51. Ordinary Least Squares (OLS)

52. OLS – Popular method of Linear Regression
• Linear relationship between dependent and independent variables
• Assumptions
– Independent variable uncorrelated with error term
– Expected value of error term is zero
– Variance of error term is constant
– Error term is normally distributed
Error
Term
Independent
Variable
Slope
Coefficient
Intercept
Dependent
Variable
Sum of Squared
Residuals (SSR)
Explained Sum
of Squares (ESS)
Total …
(TSS)
Analysis of Variance (ANOVA)

53. Coefficient of Determination – R2
• Measures percentage of total variation in dependent Y variable explained
by independent X variable. An R2 of 0.25 means X explains 25% of the
variation of Y
• R2 is equal to the correlation coefficient if there is only one independent
variable X.
• Adjusted R2 accounts for the “cost” of adding more independent variables
to the regression

54. Regression Coefficient t‐Test
• Test of statistical significance
• Use t‐test with n‐2 degrees of freedom
• Example: Test statistical significant of slope coefficient from stock return
regression below at 5% significance level, assuming standard error is 0.17.
– t‐stat = 0.9 / .17 = 5.3
– Critical t‐value = 2.2 (5%, two‐tailed, df = 10)
• Decision: Reject H0, slope coefficient
significantly different from zero

55. Confidence Intervals
β1 +/‐ tc x SE
• tc = Critical t‐value using two‐tails
with n‐2 degrees of freedom
• SE = Standard Error as previously
defined
Y +/‐ tc x sf
• tc = is same as for coefficient
• sf = Standard Error of Forecast
2 1
1
1
Coefficient Predicted Value
• Standard Error of the Regression (SER) = Standard deviation of the error terms of
the regression. Measures the degree of variability of the actual Y‐values relative to
the estimated Y‐values. The smaller the SER the greater the accuracy
• s2 = Variance of independent variable X

56. Gauss‐Markov Theorem
• If the linear regression model assumptions are true and the
regression errors display homoskedasticity, then the OLS
estimator is said to be the Best Linear Unbiased Estimator
(BLUE). This means that OLS has the following four properties:
1. Estimated coefficients have the minimum variance
compared to other methods of estimating the
coefficients
2. Estimated coefficients are based on linear functions
3. Estimated coefficients are unbiased
4. Estimate of the variance of the errors is unbiased

57. Multiple Regression
• Basic Idea: More than one independent variable
• Assumptions of multiple regression:
– Linear relationship between Y and Xs
– No exact linear relationship among Xs (related to
multicollinearity)
– Expected value of error term = 0
– Variance of error term is constant
– The model is correctly specified (ex. correct transformations for
Xs, no omitted variables)

58. Multicollinearity
• Multicollinearity occurs when two or more X variables are highly
correlated with each other.
• Effects:
– Inflated standard errors, reduces t‐stats
– Fail to reject null hypothesis too often (Type II Error)
– Variables incorrectly look unimportant
• Detection:
– Significant F‐stat overall but insignificant t‐stats
– High correlation between X variables (if only two Xs). If more than two Xs, low
correlations alone cannot rule out multicolinearilty because linear
combinations may still be highly correlated.
• Correcting for multicollinearity is typically accomplished by omitting one
or more independent variables. However, choosing the correct one(s) to
omit can be challenging. Stepwise regression is one commonly used
method.

59. Omitted Variable Bias
• Omitted Variable Bias can produce biased
estimates. An omitted variable is:
1. Correlated with the movement of at least one
independent variable and
2. Is determinant of dependent variable

60. Model Selection Criteria
• MSE is a common metric for comparing models. A ranking of
models by MSE will be identical (but reversed) to that of R2.
• MSE does not increase with more variables, which causes
downward bias in out‐of‐sample variance making it “inconsistent”
• S2 provides a simple method for reducing this bias via a penalty.
• Akaike information criterion (AIC) and Schwartz information
criterion (SIC) provide two other methods for reducing this bias.
• Penalty factors for each bias correction method is shown below.
SIC places the
greatest penalty
while s2 places
the smallest
T = number of observations
k = number of explanatory variables

61. Time Series Concepts

62. • When the variance of the residual is NOT constant across all observations. This has
no effect on estimates but can cause artificially low standard errors.
• Unconditional heteroskedasticity occurs when the heteroskedasticity is NOT
related to the level of the dependent variable. Usually causes no major problems.
Heteroskedasticity
• Conditional heteroskedasticity
often leads to artificially low
standard errors which cause t‐
stats to be too large. This may
cause Type I Error for
coefficient significance tests:
rejection of the null
hypothesis (of no significance)
when it is actually true.

63. • Detecting heteroskedasticity is most easily accomplished using Scatter Plots which
plot of residuals against each independent variable and against time.
• Correcting for heteroskedasticity is most commonly accomplished using “Robust
Standard Errors”. These can be calculated using “White‐corrected standard errors”
in the estimation.
Heteroskedasticity
Residuals
on Y‐Axis
Independents
on X‐Axis

64. Covariance Stationarity
• Constant and finite expected value
• Constant and finite variance
• Constant and finite covariance between lags
All are examples of
non‐stationarity

65. White Noise
• Strong White Noise
– Unconditional mean and variance are constant
– Serially uncorrelated and independent
– Conditional and unconditional mean/variance are same
• White Noise Process is the same as above, but allows for serial dependence.
• Normal White Noise is strong white noise that is normally distributed.
• Testing for White Noise: A Q‐Statistic is often used to test for white noise by
evaluating the overall statistical significance of the autocorrelations. The most
common is the Ljung‐Box Q‐stat (left) where n is the sample size, is the
sample autocorrelation at lag k, and h is the number of lags being tested.
• The Box‐Pierce Q‐stat (right) is the same except that it uses a simple
summation (instead of the weighted sum above) which

66. Autocorrelation Function
• Autocorrelation Function (ACF) measures
the degree of correlation with past values
of the series as a function of the number
of periods in the past (that is, the lag τ) at
which the correlation is computed.
• ACF can be used to white noise
characterizes a series b/c white noise
should not contain autocorrelation
• Partial autocorrelation function (PACF)
gives the partial correlation of a time
series with its own lagged values,
controlling for the values of the time
series at all shorter lags. It contrasts with
the autocorrelation function, which does
not control for other lags.
Only lags 1 & 2 are
significant

67. AutoRegressive Moving Average (ARMA)
Autoregressive Model
• Modeling a series as a function of
past values
• Gradual Decay: Autocorrelation has
long memory because current y is
correlated with all previous y, albeit
with decreasing strength
• ACF will show significant lags beyond
that of PACF.
• Only stationary if ‐1 < φ < 1
Moving Average Model
• Modeling a series as a function of
past residuals
• Autocorrelation Cutoff: “Very short
memory“ because y is only correlated
with a (generally) small, number of
previous y
• PACF will show significant lags
beyond that of ACF
• Stationary for any value of ϴ
AR(1)AR(p) MA(1)MA(q)
ARMA(1,1)• An ARMA includes both AR and MA terms

68. Estimating Volatilities

69. Estimating Volatility
• Continuous compounded return S over time period “I”
• Maximum likelihood estimator of variance assuming mean
return of zero; where “m” is number of observations
• Provides method for estimating today’s variance using equal
weight on historical values

70. GARCH vs EWMA
Two methods for placing more weight on recent observations
Generalized Autoregressive Conditional
Heteroskedastic (GARCH)
• Interpretation: Variance modeled as
a function of long run average, last
squared return, and last variance
• Long run avg variance = w/(1‐α–β)
• Persistence = Sum α + β. If model is
to be stationary (with mean
reversion) persistence must be < 1.
• Estimated using MLE
• Superior to EWMA when volatility is
mean reverting (which it usually is).
Exponentially Weighted Moving Average
(EWMA)
• Interpretation: Variance modeled as a
function of only last variance and last
squared return (no long run average)
• Special case of GARCH where:
– long run average (w) = 0.
– weight on last return2 (α) = 1‐λ
– Weight on last variance (β) = λ
• Requires no estimation, just supply λ
• Superior to GARCH when GARCH
persistence is >= 1 (non‐stationary)

71. Generalized Autoregressive Conditional
Heteroskedastic (GARCH)
• .
Exponentially Weighted Moving Average
(EWMA)
• .
GARCH vs EWMA
Estimating covariance
% Change in Y
% Change in XLong Run Average Covariance

72. • Covariance matrix must be positive‐semidefinite (PS). What
does that mean?
• Two examples:
– PS
– Not PS
• Small changes to a small PS matrix will likely still be PS, but
small changes to a large PS matrix may can it to not be PS.
Covariance Consistency
Variance Covariance Matrix of
dimensions n x n
Any vector of n real numbers
Transpose of w

73. Simulation Modeling
• Simulation models use random inputs that follow probability distributions
(PD) to generate scenarios (a.k.a. trials) in order to evaluate PDs of output
• Four methods for choosing input PDs
1. Bootstrapping – Construct PD by randomly drawing from historical data
2. Parameter estimation – uses parameters to define shape of specific PD
3. Best fit technique – Find PD that best fits historical data
4. Subjective guess – Construct PD based on subjective guess
• Advantages
1. Simply complex functions b/c PD of output need not be identified
2. Create visible output PDs that result from multiple input PDs
3. Allows correlation between variables
4. Easy examination effects on output variables when changing strategies
or scenarios

74. Simulation Modeling
• Incorporating Correlations – Common Approaches
– Correlations of inputs are implicitly introduced by generating joint
scenarios of input variables
– Samples of historical data are used to define the correlations between
input variables in the model
– Correlation matrix can be specified as an input
• Accuracy
– More simulations (i.e. observations, trials) can increase accuracy (see
formula for Standard Error of the Sample Mean)
– Estimator bias can be introduced via discretization error; the practice of
breaking the simulation into fixed time periods (ex, months, years). This
can be reduced by using shorter time periods, but this also increases
cost of computation

75. Pseudorandom Numbers

76. Inverse Transform Method
• Converts a random number u that is between 0 and 1 to a number from
the inverse of the cumulative distribution function (CDF)
• For discrete distributions, the unit interval [0,1] on the y‐axis (representing
the CDF) is split into segments based on the cumulative probabilities of
the discrete variables
• For example, cumulative probabilities of 40% 75% and 100% could
correspond to values 5, 20, and 40.

77. Pseudorandom Number Generators
• Reduce the variance of an estimate if the
same sequence of random numbers is
reproduced when programming the model
• Examples:
– Midsquare technique: square the first random
number and use middle digits for the next random
number
– Congruential pseudorandom number generator:
Avoids the short cycle problem of midsquare
technique