Segment 10

Calculation Problems

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

$X = \frac1{12} \sum_i^{12} X_i$

$\mu(X_i) = \frac12, \sigma^2(X_i) = 0$

$\Phi(X) = \prod_i^{12}\Phi(\frac{X_i}{12})$

$\mu(X) = \frac12 , \sigma^2(X) = \frac1{24^2}$

$S = (X \cdot 12) -6$

$\mu(S) = (12 \cdot \frac12 ) -6 = 0$

$\sigma^2(S) = (12^2 \cdot \sigma^2(X) ) - (6 \cdot \frac1{24}) = 1$

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: $\frac{4}{x^2 +4}, \frac{9}{x^3 +9}, \frac{16}{x^ +16}, \frac{25}{x^5 +25}, \frac{36}{x^6 +36}, \frac{49}{x^7 +49}, \frac{64}{x^8+64}, \frac{81}{x^9+81}, \frac{100}{x^10+100}, \frac{121}{x^11+121}$

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.