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Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
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Lecture
Exam 2 Review
STAT 225 Introduction to Probability Models March 5, 2014
Whitney Huang Purdue University
Exam 2 Review
Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
Agenda
(^1) Discrete Random Variables
(^2) Choosing a Discrete Distribution
(^3) Approximation for Discrete Distributions
(^4) Nested Problems
Notes
Notes
Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
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Bernoulli random variable
Characteristics of the Bernoulli random variable: Let X be a Bernoulli r.v. The definition of X : It is the number of success in a single trial of a random experiment The support (possible values for X ): 0: “failure" or 1: “success" Its parameter(s) and definition(s): p: the probability of success on 1 trial The probability mass function (pmf): pX (x) = px^ ( 1 − p)^1 −x^ , x = 0 , 1 The expected value: E [X ] = p The variance: Var (X ) = p( 1 − p)
Exam 2 Review
Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
Binomial random variable
Characteristics of the Binomial random variable: Let X be a Binomial r.v. The definition of X : It is the number of successes in n trials of a random experiment, where sampling is done with replacement (or trials are independent) The support: 0, 1 , · · · , n Its parameter(s) and definition(s): p: the probability of success on 1 trial and n is the sample size The probability mass function (pmf): pX (x) =
(n x
px^ ( 1 − p)n−x^ , x = 0 , 1 , · · · , n The expected value: E [X ] = np The variance: Var (X ) = np( 1 − p)
Notes
Notes
Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
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Geometric random variable:
Characteristics of the Geometric random variable: Let X be a Geometric r.v. The definition of X : The number of trials it takes to get the 1st success The support: x = 1 , 2 , · · · Its parameter(s) and definition(s): p: the probability of success in a single trial The probability mass function (pmf): pX (x) = p( 1 − p)x−^1 for x = 1 , 2 , · · · The expected value: E [X ] = (^1) p The variance: Var (X ) = (^1) p− 2 p
Properties of Geometric distribution Tail Probability: P (X > x) = ( 1 − p)x Memoryless Property: P (X > s + t|X > s) = P (X > t)
Exam 2 Review
Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
Negative Binomial random variable:
Characteristics of the Negative Binomial random variable: Let X be a Negative Binomial r.v. The definition of X : The number of trials it takes to get the rst success The support: x = r , r + 1 , r + 2 , · · · Its parameter(s) and definition(s): r : the number of success of interest p: the probability of success in a single trial The probability mass function (pmf): pX (x) =
(x− 1 r − 1
pr^ ( 1 − p)x−r^ for x = r , r + 1 , r + 2 , · · · The expected value: E [X ] = rp The variance: Var (X ) = r^ (^1 p− 2 p)
Notes
Notes
Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
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Choosing a Discrete Distribution
Exam 2 Review
Discrete Random Variables Choosing a Discrete Distribution Approximation for Discrete Distributions Nested Problems
Approximation for Discrete Distributions
Binomial approximation for Hypergeometric
If X ∼ Hyp(N, n, K ) and N is sufficiently larger than n, say N > 20 n, then we can approximate the distribution X by using X ∗^ ∼ Bin(n, p), where p = KN
Poisson approximation to Binomial
If X ∼ Binomial(n, p) with n > 100 and p < .01 then we can approximate the distribution X by using X ∗^ ∼ Poisson(λ = n × p)
Notes
Notes
