Math 141. Lecture 7: Variance, Covariance, and Sums. Albyn Jones 1. 1 Library jones/courses/141


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1 Math 141 Lecture 7: Variance, Covariance, and Sums Albyn Jones 1 1 Library jones/courses/141
2 Last Time Variance: expected squared deviation from the mean: Standard Deviation: Properties: Var(X) = E(X µ x ) 2 = σ 2 SD(X) = Var(X) = σ Var(a + bx) = b 2 Var(X) SD(a + bx) = bsd(x) Interpretation: For a symmetric distribution, the standard deviation may be considered a typical deviation from the mean. For a strongly asymmetrical distribution, it is better to work with quantiles (percentiles of the distribution).
3 Covariance and Correlation Definition: Covariance Let X and Y be two RV s with means µ x and µ y, respectively. Covariance is the expected value of the products of deviations from the means: Cov(X, Y ) = E((X µ x )(Y µ y )) Definition: Correlation Correlation is just scale free covariance: Cor(X, Y ) = ρ xy = Cov(X, Y ) σ x σ y = Cov( X σ x, Y σ y )
4 Covariance and Correlation Extreme Cases: Covariance If X = Y, i.e. the two RV s are copies of each other, then Cov(X, Y ) = Cov(X, X) = E((X µ x )(X µ x )) = Var(X) If X = Y, then Cov(X, Y ) = Cov(X, X) = Var(X) Extreme Cases: Correlation If X = Y, i.e. the two RV s are copies of each other, then If X = Y, then Cor(X, Y ) = Cor(X, Y ) = Cov(X, X) σ x σ x Cov(X, X) σ 2 x = Var(X) σ 2 x = Var(X) σ 2 x = 1 = 1
5 Interpretation Linear Association Covariance and Correlation are measures of the degree of linear association. Independent RV s have correlation 0, but correlation 0 does not guarantee independence!
6 Example Positive Covariance: linear association with positive slope. Cov(X, Y ) > 0
7 Example Negative Covariance: linear association with negative slope. Cov(X, Y ) < 0
8 Example Zero Covariance: no linear association! Cov(X, Y ) = 0
9 The variance of a sum The expected value of a sum is the sum of the expected values: E(X + Y ) = E(X) + E(Y ). What about variances? Var(X + Y ) = E((X + Y ) (µ x + µ y )) 2 Collect the terms involving X and the terms involving Y : E((X µ x ) + (Y µ y )) 2 Recalling that (a + b) 2 = a 2 + 2ab + b 2, we have E((X µ x ) 2 + (Y µ y ) 2 + 2(X µ x )(Y µ y )) Finally, the expected value of a sum is the sum of the expected values: Var(X + Y ) = E(X µ x ) 2 + E(Y µ y ) 2 + 2E ( (X µ x )(Y µ y ) )
10 The variance of a sum, part two We have produced an expression for the variance of a sum: Var(X + Y ) = E(X µ x ) 2 + E(Y µ y ) 2 + 2E ( (X µ x )(Y µ y ) ) The first two terms are Var(X) and Var(Y ), the third is 2Cov(X, Y ). Thus Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y )
11 The variance of a sum, part three Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y ) If Cov(X, Y ) > 0, then Var(X + Y ) > Var(X) + Var(Y ) If Cov(X, Y ) < 0, then Var(X + Y ) < Var(X) + Var(Y )
12 The variance of a sum: Independence Fact: If two RV s are independent, we can t predict one using the other, so there is no linear association, and their covariance is 0. Theorem: If X and Y are independent, then Var(X + Y ) = Var(X) + Var(Y ) In other words, if the random variables are independent, then the variance of their sum is the sum of their variances! Remember Pythagorus? For independent RV s, the SD of a sum works like Euclidean distance for right triangles! If Z = X + Y σ Z = σx 2 + σy 2
13 Pythagorus σ 2 X + σ2 Y σ Y σ X For the mathematically inclined, covariance is an inner product on the infinite dimensional vector space of random variables with finite variance.
14 Example: Variance of a Binomial RV Let X be a Binomial(n,p) RV. We know E(X) = np. We also know that X is the sum of n independent Bernoulli(p) trials Y i, each with E(Y i ) = p, and Var(Y i ) = pq where q = (1 p) so Var(X) = Var(Y 1 + Y Y n ) The Y s are independent, so their variances add: Var(X) = Var(Y 1 ) + Var(Y 2 ) Var(Y n ) = npq Hence we have shown Var(X) = n ( ) n (k np) 2 p k q n k = npq k k=0
15 Example: 100 coin tosses Toss a fair coin independently 100 times, and let X be the number of Heads we get. What is the appropriate probability model? X Binomial(100, 1/2) What is the expected number of heads? What is σ 2 x, the variance of X? What is σ x? What are typical outcomes of this experiment?
16 Example: 1000 coin tosses Toss a fair coin independently 1000 times, and let X be the number of Heads we get. X Binomial(1000, 1/2) What is the expected number of heads? What is σ 2 x, the variance of X? What is σ x? What are typical outcomes of this experiment?
17 Variance of a Poisson Suppose X Poisson(µ). What are E(X) and Var(X)? One can compute them directly from the definition, by summing the infinite series E(X) = k µ k e µ /k! = µ and then Var(X) = (k µ) 2 µ k e µ /k! Summing those series makes use of ideas from Math 112. Can we guess the answer without using Math 112?
18 Variance of a Poisson, Heuristically Let X be a Binomial(n,p) RV, where n is large and p is small. We know E(X) = np, and Var(X) = np(1 p). If p is small, (1 p) is close to 1, and so Var(X) np = E(X). Since that Binomial X is approximately a Poisson with parameter µ = np, we should guess for X Poisson(µ): E(X) = µ and Var(X) = µ
19 Summary Variance of a sum of two RV s: Var(X + Y ) = Var(X) + Var(Y ) + 2Cov(X, Y ) Variance of a sum of independent RV s: Var(X + Y ) = Var(X) + Var(Y ) Pythagorus and the Standard Deviation of a sum of independent RV s: SD(X + Y ) = σx 2 + σy 2 Variance of X Binomial(n, p): Var(X) = npq = np(1 p)
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