Understanding Expected Values in Statistics
Explore the concept of expected values in statistics through examples and comparisons between discrete and continuous variables. Delve into functions, variances, and more to enhance your understanding of this fundamental statistical concept.
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Presentation Transcript
Chapter 28: Expected values http://www.qualitydigest.com/inside/quality-insider-article /problems-skewness-and-kurtosis-part-one.html#
Comparison of Expected Values Discrete Continuous ? ? = ???(?) ? ? = ???? ?? ?
Example: Expected Value (class) What is the expected value in each of the following situations: a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time. 1 8 3x 8 0 + + 0 x 2 = = 2 0 8 x 8.5 f (x) = = f (x) X X else else
Chapter 29: Functions, Variance http://quantivity.files.wordpress.com/2011/05/us-sector-2004-empiricals1.png?w=600h=396 http://quantivity.wordpress.com/2011/05/02/empirical-distribution-minimum-variance/
Comparison of Functions, Variances Discrete ? ?(?) Continuous ? ?(?) ?(?)??? ?? Function (general) = ?(?)??(?) = ? Function (X2) ? ?2= ?2??(?) ? ?2= ?2??? ?? ? Variance Var(X) = ?(X2) (?(X))2 Var(X) = ?(X2) (?(X))2 SD ??= ???(?) ??= ???(?)
Example: Expected Value - function (class) What is ?(X2) in each of the following situations: a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time. 1 8 3x 8 0 + + 0 x 2 = = 2 0 8 x 8.5 f (x) = = f (x) X X else else
Example: Variance (class) What is the variance in each of the following situations: a) The following is the density of the magnitude X of a dynamic load on a bridge (in newtons) b) The train to Chicago leaves Lafayette at a random time between 8 am and 8:30 am. Let X be the departure time. 1 8 3x 8 0 + + 0 x 2 = = 2 0 8 x 8.5 f (x) = = f (x) X X else else
Friendly Facts about Continuous Random Variables - 1 Theorem 28.18: Expected value of a linear sum of two or more continuous random variables: ?(a1X1+ + anXn) = a1?(X1) + + an?(Xn) Theorem 28.19: Expected value of the product of functions of independent continuous random variables: ?(g(X)h(Y)) = ?(g(X))?(h(Y))
Friendly Facts about Continuous Random Variables - 2 Theorem 28.21: Variances of a linear sum of two or more independent continuous random variables: Var(a1X1+ + anXn) =?1 Corollary 28.22: Variance of a linear function of continuous random variables: Var(aX + b) = a2Var(X) 2Var(X1) + + ?? 2Var(Xn)
