birth order associated with increased risk of downs syndrome, etc…

Lecture 9

The Sharpe ratio

\(S(X) = \frac{\mu_X - r_f}{\sigma_X}\) - higher better - how to compute with a calculator from a pdf and also with just \(\mu\) and \(\sigma\) - computing in R

Joint pdfs

Find probability given values Are they independent or not? What do increasing and decreasing joint pdfs look like?

\(E(X+Y) = E(X) + E(Y)\) regardless of independence

3 Rules of independence

Covariance of RV

\(Cov(X, Y) = E((X - \mu_X)(Y - \mu_Y))\)\(Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)\) If \(X\),\(Y\) are independent, then \(Cov(X, Y) = 0\), opposite not true If \(X\), \(Y\) are independent, \(Cov(X, Y) = 0\), so \(Var(X+ Y) = Var(X) + Var(Y)\)

Correlation

\(corr(x, y) = \frac{cov(x, y)}{s_x \cdot s_y}\)\[\rho = Corr(X, Y) = \frac{Cov(X, Y)}{\sigma_X \cdot \sigma_Y}\] ## IID variables \(E(aX + bY + c) = aE(X) + bE(Y) + c\)\(Var(aX + bY + c) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y)\)

Lecture 10

Bernoulli trial

expectation and variance

Binomial

\(Y = B_1 + B_2 + ... + B_n\), where \(B_1, B_2, ..., B_n\) Bernoulli trials FIST expectation and variance binomial pdf limiting of \(p\) approaches Poisson distribution use R to find these probabilities

Poisson

RIPS expectation and variance use R to find these probabilities

Lecture 11

As \(n\) increases binomial approaches normal distribution

Shifts and scales of normal distribution

Z score and standardization Use symmetry to find probabilities, finding them in R Percentiles to Z values, finding them in R