Segment 10

From Computational Statistics (CSE383M and CS395T)
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Calculation Problems

1. 12 random values between 0 to 1. Add them up and subtract 6.

<math> X = \frac1{12} \sum_i^{12} X_i</math>

<math> \mu(X_i) = \frac12, \sigma^2(X_i) = 0 </math>

<math>\Phi(X) = \prod_i^{12}\Phi(\frac{X_i}{12})</math>

<math> \mu(X) = \frac12 , \sigma^2(X) = \frac1{24^2} </math>

<math> S = (X \cdot 12) -6 </math>

<math> \mu(S) = (12 \cdot \frac12 ) -6 = 0 </math>

<math> \sigma^2(S) = (12^2 \cdot \sigma^2(X) ) - (6 \cdot \frac1{24}) = 1 </math>

2. Invent a family of functions that have value 1 at x=0, and have zero derivative at x = 0 and reach 0 at large x. Multiply 10 of them and graph the result.

The functions: <math> \frac{4}{x^2 +4}, \frac{9}{x^3 +9}, \frac{16}{x^ +16}, \frac{25}{x^5 +25}, \frac{36}{x^6 +36}, \frac{49}{x^7 +49}, \frac{64}{x^8+64}, \frac{81}{x^9+81}, \frac{100}{x^10+100}, \frac{121}{x^11+121} </math>

Class Activity

Solutions found in Jin teamed with Rcardenas, Sillu, Travis, Jin

Notes to self

  1. Characteristic function of a Gaussian of a Normal is Normal
  2. Cauchy distribution doesn't have convergent Taylor series there for doesn't apply for Cauchy
  3. Conditions: N has to be large, higher moments have to be well behaved, Taylor series expansion exists.