Introduction Some Basic Statistical Concepts Some Useful Discrete Distributions Some Useful Continuous Distributions Ch1. Review of Basic Probability and Statistics Terminology Zhang Jin-Ting Department of Statistics and Applied Probability July 16, 2012 Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
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Ch1. Review of Basic Probability and Statistics Terminology
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IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Ch1. Review of Basic Probability andStatistics Terminology
Zhang Jin-Ting
Department of Statistics and Applied Probability
July 16, 2012
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Outline
Introduction
Some Basic Statistical Concepts
Some Useful Discrete Distributions
Some Useful Continuous Distributions
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
AimsI This chapter is a very brief review of basic properties and
terminology from probability and statistics that we will usein this semester.
I The student is assumed to have seen most of this chapterbefore.
I This material is treated in most introductory texts onprobability and statistics.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
AimsI This chapter is a very brief review of basic properties and
terminology from probability and statistics that we will usein this semester.
I The student is assumed to have seen most of this chapterbefore.
I This material is treated in most introductory texts onprobability and statistics.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
AimsI This chapter is a very brief review of basic properties and
terminology from probability and statistics that we will usein this semester.
I The student is assumed to have seen most of this chapterbefore.
I This material is treated in most introductory texts onprobability and statistics.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Cumulative Distribution FunctionI Given a real-valued random variable X , the Cumulative
Distribution Function (c.d.f.) of X is the functionF : R → [0, 1] defined by
F (x) = Pr(X ≤ x), x ∈ R.
I If there exists a function f : R → [0,∞) such thatF (x) =
∫ x−∞ f (y)dy for every x ∈ R, then x is said to be
continuous with Probability Density Function (p.d.f.) f .
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Cumulative Distribution FunctionI Given a real-valued random variable X , the Cumulative
Distribution Function (c.d.f.) of X is the functionF : R → [0, 1] defined by
F (x) = Pr(X ≤ x), x ∈ R.
I If there exists a function f : R → [0,∞) such thatF (x) =
∫ x−∞ f (y)dy for every x ∈ R, then x is said to be
continuous with Probability Density Function (p.d.f.) f .
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Probability Mass FunctionI On the other hand, if X only takes values in the set of
integers, or more generally in some countable (or finite) setS, then its c.d.f. is completely determined by its ProbabilityMass Function (p.m.f.), p : S → [0, 1] where
pi = Pr(X = i), i ∈ S.
I Clearly,∫∞−∞ f (x)dx = 1 for any p.d.f. and
∑i∈S pi = 1 for
any p.m.f.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Probability Mass FunctionI On the other hand, if X only takes values in the set of
integers, or more generally in some countable (or finite) setS, then its c.d.f. is completely determined by its ProbabilityMass Function (p.m.f.), p : S → [0, 1] where
pi = Pr(X = i), i ∈ S.
I Clearly,∫∞−∞ f (x)dx = 1 for any p.d.f. and
∑i∈S pi = 1 for
any p.m.f.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Mean and ExpectationI The Mean or Expectation of a real-valued random variable
X is defined by
E(X ) =
{∫∞−∞ xf (x)dx if X has p.d.f. f∑
i∈S ipi if X has p.m.f. p.
I The Variance of X , denoted by var(X ), is defined to beE [(X − E(X ))2] and equals E(X 2)− [E(X )]2 when it isfinite. If we view X as the value of some measurement,then the standard deviation
√var(X ) determines the
magnitude of error in this measurement.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Mean and ExpectationI The Mean or Expectation of a real-valued random variable
X is defined by
E(X ) =
{∫∞−∞ xf (x)dx if X has p.d.f. f∑
i∈S ipi if X has p.m.f. p.
I The Variance of X , denoted by var(X ), is defined to beE [(X − E(X ))2] and equals E(X 2)− [E(X )]2 when it isfinite. If we view X as the value of some measurement,then the standard deviation
√var(X ) determines the
magnitude of error in this measurement.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Sample MeanI Let X1, X2, · · · be a sequence of Independent Identically
Distributed (iid) random variables with mean µ andvariance σ2. We define the n-th Sample Mean by
X̄n =1n
n∑
i=1
Xi .
I Then E(X̄n) = µ and var(X̄n) = σ2
n .
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Sample MeanI Let X1, X2, · · · be a sequence of Independent Identically
Distributed (iid) random variables with mean µ andvariance σ2. We define the n-th Sample Mean by
X̄n =1n
n∑
i=1
Xi .
I Then E(X̄n) = µ and var(X̄n) = σ2
n .
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
CLT and CII When n is sufficiently large, the Central Limit Theorem
implies X̄n−µσ/√
n has an approximate N(0, 1) distribution.Therefore,
Pr(X̄n − 1.96σ√n≤ µ ≤ X̄n + 1.96
σ√n
) ≈ 0.95
for sufficiently large n.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Sample VarianceI When σ2 is unknown, its value can be estimated by the
Sample Variance
S2n =
1n − 1
n∑
i=1
(Xi − X̄n)2.
I The sample variance is an Unbiased Estimator of σ2
whenever the Xi ’s are iid.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
Sample VarianceI When σ2 is unknown, its value can be estimated by the
Sample Variance
S2n =
1n − 1
n∑
i=1
(Xi − X̄n)2.
I The sample variance is an Unbiased Estimator of σ2
whenever the Xi ’s are iid.
Zhang J.T. Ch1. Review of Basic Probability and Statistics Terminology
IntroductionSome Basic Statistical Concepts
Some Useful Discrete DistributionsSome Useful Continuous Distributions
CovarianceI Consider two random variables X and Y (have some Joint
Distribution). The Covariance of X and Y is defined by