ISYE6644: Module 7 (Random
Variate Generation) with
complete verified solutions
Unif(0, 1)
__________ PRNs are used to generate observations (variates) from
other distributions, and even stochastic processes.
Unif(0, 1)
By the Inverse Transform Theorem, if X is a continuous random
variable with cdf F(x), then the random variable F(X) ~ _________.
X
Let U ~ Unif(0, 1). Then F(X) = U means that the random variable
F^(-1)(U) has the same distribution as _____.
DISC; CONT
The Arena functions _________ and ________ can be used to generate
random variables from the empirical cdf's of discrete and
continuous distributions, respectively.
Glivenko-Cantelli
The ________________ Lemma says that the empirical cdf approaches a
true cdf F(x) for all x as n approaches infinity.
Empirical
If you can't find a good theoretical distribution to model a certain
random variable, you may want to use the _____________ cdf of the
data X_1, X_2, ..., X_n.
1/n
The empirical cdf of the data X_1, X_2, ..., X_n is a step function with
jumps of _______ (every time an observation occurs).
Convolution
______________ refers to adding things up.
Tria(0,1,2)
, If U_1 and U_2 are i.i.d. Unif(0, 1), then U_1 + U_2 is ___________.
Erlang_n(lambda)
The sum of n i.i.d. Exp(lambda) random variables are ____________.
NegBin(n, p)
The sum of n i.i.d. Geom(p) random variables are __________.
Chi-Squared; degrees of freedom
The sum of n standard normal random variables are ___________ with
n ______________ _____ ____________.
Cauchy
The sample mean of n Cauchy random variables is ____________.
Acceptance-Rejection
The ___________________ method samples from a distribution that is
"almost" the one we want, and then adjusts by "accepting" only a
certain proportion of those samples.
Composition
The ______________ Algorithm:
1. Generate a positive integer J such that P( J = j) = p_j for all j.
2. Return X from cdf F_J(x).
Acceptance-Rejection
The ____________________ Algorithm for the continuous case:
1. Generate U from Unif(0, 1).
2. Generate Y from h(y) = t(y)/c (independent of U) until U <= g(Y) =
f(Y)/t(Y) = f(Y)/[ch(Y)].
e. Return X <- Y.
N(0, 1)
The Box-Muller Method generates two i.i.d. __________ random
variables from two i.i.d. Unif(0,1) random variables.
Cauchy; t(1)
Variate Generation) with
complete verified solutions
Unif(0, 1)
__________ PRNs are used to generate observations (variates) from
other distributions, and even stochastic processes.
Unif(0, 1)
By the Inverse Transform Theorem, if X is a continuous random
variable with cdf F(x), then the random variable F(X) ~ _________.
X
Let U ~ Unif(0, 1). Then F(X) = U means that the random variable
F^(-1)(U) has the same distribution as _____.
DISC; CONT
The Arena functions _________ and ________ can be used to generate
random variables from the empirical cdf's of discrete and
continuous distributions, respectively.
Glivenko-Cantelli
The ________________ Lemma says that the empirical cdf approaches a
true cdf F(x) for all x as n approaches infinity.
Empirical
If you can't find a good theoretical distribution to model a certain
random variable, you may want to use the _____________ cdf of the
data X_1, X_2, ..., X_n.
1/n
The empirical cdf of the data X_1, X_2, ..., X_n is a step function with
jumps of _______ (every time an observation occurs).
Convolution
______________ refers to adding things up.
Tria(0,1,2)
, If U_1 and U_2 are i.i.d. Unif(0, 1), then U_1 + U_2 is ___________.
Erlang_n(lambda)
The sum of n i.i.d. Exp(lambda) random variables are ____________.
NegBin(n, p)
The sum of n i.i.d. Geom(p) random variables are __________.
Chi-Squared; degrees of freedom
The sum of n standard normal random variables are ___________ with
n ______________ _____ ____________.
Cauchy
The sample mean of n Cauchy random variables is ____________.
Acceptance-Rejection
The ___________________ method samples from a distribution that is
"almost" the one we want, and then adjusts by "accepting" only a
certain proportion of those samples.
Composition
The ______________ Algorithm:
1. Generate a positive integer J such that P( J = j) = p_j for all j.
2. Return X from cdf F_J(x).
Acceptance-Rejection
The ____________________ Algorithm for the continuous case:
1. Generate U from Unif(0, 1).
2. Generate Y from h(y) = t(y)/c (independent of U) until U <= g(Y) =
f(Y)/t(Y) = f(Y)/[ch(Y)].
e. Return X <- Y.
N(0, 1)
The Box-Muller Method generates two i.i.d. __________ random
variables from two i.i.d. Unif(0,1) random variables.
Cauchy; t(1)
