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Unformatted text preview: omes, where the weights are the probabilities of outcomes. Discrete case: Suppose X can take n possible values, denoted as x1, x2, …, xn (keep in mind the notational convention). Let pi = prob (X = xi), ∑ p 1, and p 0 for any . Then μ ≡ E ( X ) = ∑ i =1 pi xi = p1 x1 + p2 x2 + ... + pn xn , where E() is the expectation n operator, and μ is used to denote a mean. Example: Flip a coin, if head you get 1 dollar and if tail you get nothing. Suppose it is a fair coin, then you are expected to get 0.5 dollars, that is, the expected value of flipping a coin is 0.5 dollars, although if you flip a coin another time you could get 1 dollar or nothing. Facts about expected values 1. For any constant a, E(a) = a. A constant does not have any variations! Example: E(3)=3 2. For any constants a and b, E(a + bX) = a + bE(X) Example: In the previous GPA and study time example, the estimated regression line is Y=1+0.1X, where Y=GPA, X=study time per week measured in hours. Suppose the population mean of study time is 15 hours per week, that is, E(X)=15, then GPA is expected to be E(Y) = 1+0.1 E(X) = 2.5 3. For any constants a , a , … , a , and r.v.’s X , X , … , X , ∑ ∑ . For example, E(X1+X2+…+Xn)=E(X1)+E(X2)+…+E(Xn). That is, the expected value of the sum is the sum of expected values. 3 If Y = 1 + 2 X + u , E(u)=0, then E(Y)=? 4. Note that the above rule only applies to a linear form, so E[(αX)2]=α2E(X2) ≠ α2[E(X)]2. Example: Running business: X=profit, three possible outcomes, x1=1 m, p1=0.3; x2=‐1.5m, p2=0.3; x3=0.1m, p3=0.4 The r.v. X2=1 m2, p1=0.3; X2=2.25 m2,p2=0.3; X2=0.01 m2, p3=0.4 Then E(X2) =0.3*1 +0.3*2.25 +0.3*0.01 =0.978 [E(X)]2=(‐0.11m)^2=0.0121 m2 So E(X2)≠ [E(X)]2 Variance A measure of variability or dispersion of a r.v. around its mean, or how spread out the distribution is. Figure: distributions of r.v.’s with different variances Var ( X ) = σ X 2 = E ( X − μ X )2 = E ( X 2 − 2 X μ X + μ X 2 ) = E ( X 2 ) − μ X 2 , where Facts about variances 1. Positive square root of a variance is a standard deviation (sd), or σ X . 2. Var and sd are always nonnegative. 3....
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