Example: X = Cholesterol level (mg/dL)

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Presentation transcript:

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Suppose X is transformed to another random variable, say h(X). Then by def,

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Suppose X is constant, say b, throughout entire population… b Then by def,

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Suppose X is constant, say b, throughout entire population… Then…

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… a Then by def,

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… Then… i.e.,…

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… Add any constant b to X… Then… i.e.,…

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… Add any constant b to X… Then… i.e.,…

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… then X is also multiplied by a.

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… then X is also multiplied by a. i.e.,… i.e.,…

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Add any constant b to X… then b is also added to X .

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Add any constant b to X… then b is also added to X . i.e.,… i.e.,…

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X This is the analogue of the “alternate computational formula” for the sample variance s2.

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Multiply X by any constant a… Add any constant b to X… Then… i.e.,…

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Example: X = Cholesterol level (mg/dL) random variable X Discrete General Properties of “Expectation” of X Add any constant b to X… then b is also added to X . i.e.,… i.e.,…

Random Variable Y = “Sum” Example: Roll two dice, independently. Sample space {(1,1),…,(6,6)} Discrete Random Variable Y = “Sum” 0.14 0.16 0.20 0.30 0.15 0.10 0.05 P(Y = y) y p (y) 2 0.028 3 0.074 4 0.116 5 0.153 6 0.170 7 0.169 8 0.132 9 0.082 10 0.047 11 0.022 12 0.007 1 pmf pmf y = 2:12 p = c(.028,.074, .116,.153,.170, .169,.132,.082, .047,.022,.007) mu = sum(y*p) sig.sqd = sum((y-mu)^2*p) print(c(mu, sig.sqd)) x p1 (x) p2 (x) 1 0.20 0.14 2 0.30 0.16 3 4 0.15 5 0.10 6 0.05

Example: X = Cholesterol level (mg/dL) POPULATION Pop values x Probabilities pmf p(x) x1 p(x1) x2 p(x2) x3 p(x3) ⋮ Total 1 Discrete random variable X Example: X = Cholesterol level (mg/dL) General Properties of “Expectation” of X Theorem Theorem If X and Y are independent, then These formulas apply to both discrete and continuous random variables. These will be proved in Chapter 8 (Multivariate setting),