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Outline Random Variables - Variance K. Subramani1 1 Lane Department of Computer Science and Electrical Engineering West Virginia University 26 January, 2012 Subramani Probability Theory Outline Outline 1 Recap Subramani Probability Theory Outline Outline 1 Recap 2 Variance Subramani Probability Theory Outline Outline 1 Recap 2 Variance 3 Covariance Subramani Probability Theory Outline Outline 1 Recap 2 Variance 3 Covariance 4 Identities Subramani Probability Theory Outline Outline 1 Recap 2 Variance 3 Covariance 4 Identities 5 Variance of some common random variables Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Recap Main points Random Variable, Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Recap Main points Random Variable, Distribution of a random variable (pmf), Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Recap Main points Random Variable, Distribution of a random variable (pmf), Expected Value. Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Variance Note The variance of a random variable measures the spread of its distribution. Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Variance Note The variance of a random variable measures the spread of its distribution. In many respects, it is the dual of its expectation, which is actually a clustering measure. Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Variance Note The variance of a random variable measures the spread of its distribution. In many respects, it is the dual of its expectation, which is actually a clustering measure. Definition Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Variance Note The variance of a random variable measures the spread of its distribution. In many respects, it is the dual of its expectation, which is actually a clustering measure. Definition Let X denote a random variable. Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Variance Note The variance of a random variable measures the spread of its distribution. In many respects, it is the dual of its expectation, which is actually a clustering measure. Definition Let X denote a random variable. The variance of X , denoted by Var[X ] is defined as: E[(X − E[X ])2 ]. Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Variance Note The variance of a random variable measures the spread of its distribution. In many respects, it is the dual of its expectation, which is actually a clustering measure. Definition Let X denote a random variable. The variance of X , denoted by Var[X ] is defined as: E[(X − E[X ])2 ]. Note Var[X ] = E[X 2 ] − (E[X ])2 . Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Covariance Definition Let X and Y denote two random variables. Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Covariance Definition Let X and Y denote two random variables. The covariance between X and Y is defined as: Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Covariance Definition Let X and Y denote two random variables. The covariance between X and Y is defined as: Cov(X , Y ) = E[(X − E[X ]) · (Y − E[Y ])] Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Covariance Definition Let X and Y denote two random variables. The covariance between X and Y is defined as: Cov(X , Y ) = E[(X − E[X ]) · (Y − E[Y ])] Simplification Cov(X , Y ) = E[X · Y ] − E[X ] · E[Y ] Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Identities Properties of Variance and Covariance Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Identities Properties of Variance and Covariance 1 Cov(X , X ) = Var(X ). Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Identities Properties of Variance and Covariance 1 Cov(X , X ) = Var(X ). 2 Cov(X , Y ) = Cov(Y , X ). Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Identities Properties of Variance and Covariance 1 Cov(X , X ) = Var(X ). 2 Cov(X , Y ) = Cov(Y , X ). 3 Cov(c · X , Y ) = c · Cov(X , Y ). Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Identities Properties of Variance and Covariance 1 Cov(X , X ) = Var(X ). 2 Cov(X , Y ) = Cov(Y , X ). 3 Cov(c · X , Y ) = c · Cov(X , Y ). 4 Cov(X , Y + Z ) = Cov(X , Y ) + Cov(X , Z ). Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Identities Properties of Variance and Covariance 1 Cov(X , X ) = Var(X ). 2 Cov(X , Y ) = Cov(Y , X ). 3 Cov(c · X , Y ) = c · Cov(X , Y ). 4 Cov(X , Y + Z ) = Cov(X , Y ) + Cov(X , Z ). P P Var( ni=1 (Xi )) = ni=1 Var(Xi ), 5 Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Identities Properties of Variance and Covariance 1 Cov(X , X ) = Var(X ). 2 Cov(X , Y ) = Cov(Y , X ). 3 Cov(c · X , Y ) = c · Cov(X , Y ). 4 Cov(X , Y + Z ) = Cov(X , Y ) + Cov(X , Z ). P P Var( ni=1 (Xi )) = ni=1 Var(Xi ), if X1 , X2 , . . . Xn are independent random variables. 5 Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = (1 − p) Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = (1 − p) p(1) = p Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = (1 − p) p E[X ] = 0 · (1 − p) + 1 · p Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = (1 − p) p E[X ] = 0 · (1 − p) + 1 · p = p Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = (1 − p) p E[X ] = 0 · (1 − p) + 1 · p = p E[X 2 ] = Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = (1 − p) p E[X ] = 0 · (1 − p) + 1 · p = p E[X 2 ] = 12 · p Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = (1 − p) p E[X ] = 0 · (1 − p) + 1 · p = p E[X 2 ] = 12 · p + 02 · (1 − p) Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = p E[X ] = 0 · (1 − p) + 1 · p = p E[X 2 ] = 12 · p + 02 · (1 − p) = p Var[X ] (1 − p) = Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = p E[X ] = 0 · (1 − p) + 1 · p = p E[X 2 ] = 12 · p + 02 · (1 − p) = p Var[X ] (1 − p) = Subramani E[X 2 ] − (E[X ])2 Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = p E[X ] = 0 · (1 − p) + 1 · p = p E[X 2 ] = 12 · p + 02 · (1 − p) = p Var[X ] (1 − p) = E[X 2 ] − (E[X ])2 = p − p2 Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Bernoulli Variable Computation For some p, 0 ≤ p ≤ 1, p(0) = p(1) = p E[X ] = 0 · (1 − p) + 1 · p = p E[X 2 ] = 12 · p + 02 · (1 − p) = p Var[X ] (1 − p) = E[X 2 ] − (E[X ])2 = p − p2 = p · (1 − p) Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Binomial Variable Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Binomial Variable Computation Observe that if X is a binomially distributed random variable with parameters n and p, then P it can be expressed as a sum of n independent Bernoulli variables, i.e., X = ni=1 Xi , where each Xi is a Bernoulli random variable with parameter p. Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Binomial Variable Computation Observe that if X is a binomially distributed random variable with parameters n and p, then P it can be expressed as a sum of n independent Bernoulli variables, i.e., X = ni=1 Xi , where each Xi is a Bernoulli random variable with parameter p. Using the identity on the variance of a sum, we conclude that Var(X ) = Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Binomial Variable Computation Observe that if X is a binomially distributed random variable with parameters n and p, then P it can be expressed as a sum of n independent Bernoulli variables, i.e., X = ni=1 Xi , where each Xi is a Bernoulli random variable with parameter p. Using the identity on the variance of a sum, we conclude that Var(X ) = n · p · (1 − p). Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Geometric Variable Statement If X is a geometric random variable with parameter p, Subramani Probability Theory Recap Variance Covariance Identities Variance of some common random variables Geometric Variable Statement If X is a geometric random variable with parameter p, Var(X ) = Subramani 1−p p2 Probability Theory
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