Lectures on Elementary Probability

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34 CHAPTER 5. CONTINUOUS RANDOM VARIABLES 5.2 Variance The variance of X is σ 2 X = Var(X) = E[(X − µ X ) 2 ]. (5.18) The standard deviation of X is the square root of the variance. By the theorem, the variance may be computed by σ 2 X = ∫ ∞ −∞ (x − µ X ) 2 f X (x) dx. (5.19) It is thus a weighted mean of squared deviations from the mean of the original random variable. Sometimes this is called a “mean square”. Then the standard deviation is a “root mean square”. The following result is useful in some computations. Theorem 5.4 The variance is given by the alternative formula 5.3 Joint densities σ 2 X = E[X 2 ] − E[X] 2 . (5.20) Consider continuous random variables X and Y with densities f X (x) and f Y (y). They have a joint probability density function f X,Y if for every region in the plane the probability ∫ ∫ P [(X, Y ) ∈ C] = f X,Y (x, y) dx dy. (5.21) In a similar way, we can talk about the joint probability density function of three or more random variables. Given the joint probability density function of X, Y , it is easy to compute the the probability density function of X, and it is equally easy to compute the probability density function of Y . The first formula is f X (x) = ∫ ∞ −∞ C f X,Y (x, y) dy. (5.22) Similarly f Y (y) = ∫ ∞ −∞ f X,Y (x, y) dx. (5.23) Theorem 5.5 The expectation of a random variable Z = g(X, Y ) that is a function of random variables X, Y may be computed in terms of the joint probability density function of X, Y by E[g(X, Y )] = ∫ ∞ ∫ ∞ −∞ −∞ g(x, y)f X,Y (x, y) dx dy. (5.24)
5.4. UNCORRELATED RANDOM VARIABLES 35 Proof: E[g(X, Y )] = ∫ ∞ 0 Writing this in terms of the joint density gives ∫ 0 P [g(X, Y ) > z] dz − P [g(X, Y ) < z] dz. (5.25) −∞ ∫ ∞ ∫ ∫ ∫ 0 ∫ ∫ E[g(X)] = f(x, y) dx dy dz − f(x, y) dx dy dz. 0 g(x,y)>z −∞ g(x,y)0 g(x,y)>0 ∫ g(x,y) 0 ∫ ∫ ∫ 0 dzf(x, y) dx dy− dzf(x, y) dx dy. g(x,y)
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Lectures on Elementary Probability

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