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Expectation

Expectation. Expected value:. Expectation of Sum of R.V. X and Y r.v. with joint pdf p(x,y) : Example: R = the return on the S&P500 each day (continuously compounded) Distributed according to h ( r ) What is the return over two days?. Expectation of Sum of R.V. In general,

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Expectation

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  1. Expectation • Expected value:

  2. Expectation of Sum of R.V. • X and Y r.v. with joint pdf p(x,y): • Example: • R = the return on the S&P500 each day (continuously compounded) • Distributed according to h(r) • What is the return over two days?

  3. Expectation of Sum of R.V. • In general, • Consider Example 2c, p. 300 • Consider Examples 2e and 2f, p. 301

  4. Covariance • Proposition 4.1. • If X and Y are independent, then, for any functions h and g:

  5. Covariance • The covariance between X and Y, denoted byCov(X,Y) is defined by:

  6. Covariance • Proposition 4.2. (i) (ii) (iii) (iv) Example Ch7, 4a

  7. Variance • Example 4b, p. 324

  8. Conditional Expectation • Conditional pdf: • Consider Example 5b

  9. Computing Expectations by Conditioning • Proposition 5.1: • Consider Example 5d, p. 335

  10. Moment Generating Functions • M(t) = moment generation function of r.v. X if X is discrete with mass function p(x) if X is continuous with density f(x)

  11. Moment Generating Functions • M(t) gives all moments by differentiation: • Examples: M’(0), M”(0), etc. • Examples 7b, 7c, 7d

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