The R functions we have discussed in these weeks:
| functions |
| qnorm() |
| pnorm() |
| cov() |
| cor() |
| mutate() |
| View() |
| tibble() |
| slice() |
| exp() |
| lfactorial() |
| dbinom() |
| dpois() |
| slice_sample() |
| c() |
\(\mu = E(X)\) \(= x_1p(x_1) + x_2p(x_2) + ... + x_np(x_n)\)
How to define a pdf!
-\(E(X \pm c) = E(X) \pm c\)
-\(SD(X \pm c) = SD(X)\)
-\(Var(X \pm c) = Var(X)\)
-\(E(cX) = cE(X)\)
-\(SD(cX) = |c|SD(X)\)
-\(Var(cX) = c^2Var(X)\)
-\(E(cX \pm a)=cE(X) \pm a\)
- \(Var(cX \pm a)=c^2Var(X)\)
\(cov(x, y) = \frac{(x_1 - \bar{x})(y_1 - \bar{y}) + (x_2 - \bar{x})(y_2 - \bar{y}) + \ldots + (x_n - \bar{x})(y_n - \bar{y})}{n-1}\)
\[corr(x, y) = \frac{cov(x, y)}{s_x \cdot s_y}\]
\(m = \frac{r \cdot s_y}{s_x}\)
\(b = \bar{y} - m \bar{x}\)
library(tidymodels)
## Assign the least
## squares line
least_squares_fit <-
linear_reg() %>%
set_engine("lm") %>%
fit(price ~ carat, data = diamonds)
## NOTE: outcome first, predictor second
## Find the prediction
predict(least_squares_fit, tibble(carat = c(1, 2, 2.5, 4)))# A tibble: 4 × 1
.pred
<dbl>
1 5500.
2 13256.
3 17135.
4 28769.birth order associated with increased risk of downs syndrome, etc…
\(S(X) = \frac{\mu_X - r_f}{\sigma_X}\) - higher better - how to compute with a calculator from a pdf and also with just \(\mu\) and \(\sigma\) - computing in R
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
\(Cov(X, Y) = E((X - \mu_X)(Y - \mu_Y))\) \(Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y)\) If \(X\),\(Y\) are independent, then \(Cov(X, Y) = 0\), opposite not true If \(X\), \(Y\) are independent, \(Cov(X, Y) = 0\), so \(Var(X+ Y) = Var(X) + Var(Y)\)
\(corr(x, y) = \frac{cov(x, y)}{s_x \cdot s_y}\) \[\rho = Corr(X, Y) = \frac{Cov(X, Y)}{\sigma_X \cdot \sigma_Y}\] ## IID variables \(E(aX + bY + c) = aE(X) + bE(Y) + c\) \(Var(aX + bY + c) = a^2Var(X) + b^2Var(Y) + 2abCov(X, Y)\)
expectation and variance
\(Y = B_1 + B_2 + ... + B_n\), 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
RIPS expectation and variance use R to find these probabilities
As \(n\) increases binomial approaches normal distribution
Z score and standardization Use symmetry to find probabilities, finding them in R Percentiles to Z values, finding them in R
SRS, stratified, cluster, census, voluntary response, convenience benefits and drawbacks of all and where to apply each
distribution of the sample mean, standard error, control limits and types of errors, error rates with repeated testing, control charts for variation