Lectures on Elementary Probability

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20 CHAPTER 3. DISCRETE RANDOM VARIABLES Theorem 3.7 If X and Y are random variables with expectations E[X] and E[Y ], then they satisfy the addition property E[X + Y ] = E[X] + E[Y ]. (3.18) Important note: The theorem can also be stated using the other notation for mean in the form µ X+Y = µ X + µ Y . Proof: E[X + Y ] = ∑ ∑ (x + y)P [X = x, Y = y]. (3.19) x y This is equal to ∑ ∑ xP [X = x, Y = y] + ∑ x y x ∑ yP [X = x, Y = y] = E[X] + E[Y ]. (3.20) 3.4 Uncorrelated random variables The covariance of random variables X and Y is defined by In particular, the variance is given by y Cov(X, Y ) = E[(X − µ X )(Y − µ Y )]. (3.21) σ 2 X = Var(X) = Cov(X, X). (3.22) Theorem 3.8 The variance of a sum is given in terms of the variances of the terms and the covariance by Var(X + Y ) = Var(X) + Cov(X, Y ) + Var(Y ). (3.23) The correlation of random variables X and Y is defined by ρ(X, Y ) = Cov(X, Y ) σ X σ Y . (3.24) Thus the result of the theorem may also be written in the form σ 2 X+Y = σ 2 X + 2ρ(X, Y )σ X σ Y + σ 2 Y . (3.25) Random variables are said to be uncorrelated if their correlation is zero. This is the same as saying that their covariance is zero. Theorem 3.9 For uncorrelated random variables the variances add. That is, if Cov(X, Y ) = 0, then σ 2 X+Y = σ 2 X + σ 2 Y . (3.26)
3.5. INDEPENDENT RANDOM VARIABLES 21 3.5 Independent random variables Consider discrete random variables X and Y . They are said to be independent if the events X = x, Y = y are independent for all possible values x, y. That is, X, Y are independent if P [X = x, Y = y] = P [X = x][P [Y = y] (3.27) for all possible values x, y of the two random variables. There is of course an appropriate generalization of the notion of independence to more than two random variables. Again there is a multiplication rule, but this time with more than two factors. Theorem 3.10 If X and Y are independent random variables, and f and g are arbitrary functions, then the random variables f(X) and g(Y ) satisfy the multiplication property E[f(X)g(Y )] = E[f(X)]E[g(Y )]. (3.28) This theorem has an important consequence: Independent random variables are uncorrelated. It immediately follows that for independent random variables the variances add. Theorem 3.11 If X and Y are independent random variables, then X and Y are uncorrelated random variables. Proof: If X and Y are independent, then from the previous theorem E[(X − µ X )(Y − µ Y )] = E[X − µ X ]E[Y − µ Y ] = 0 · 0 = 0. (3.29)
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