• The mutually exclusive results of a random process are called the outcomes - only one of the possible outcomes can be observed.

• Probability of an outcome is the proportion that the outcome occurs in the long run (if the experiment is repeated many times.)

• The set of all possible outcomes of a random variable is called the sample space.

• An event is a subset of the sample space and consists of one or more outcomes.

• Random variables can be discrete (0 and 1) or continuous.

Example: rolling a dice once.

• This is nothing but randomly selecting a sample of size 1 from a set of numbers which are mutually exclusive outcomes.

• The sample space is {1,2,3,4,5,6}, and we can think of many different events, e.g., ’the observed outcome lies between 1 and 3.

• Let’s role a dice once: ###

sample(1:6, 1) 

## [1] 2

• The probability distribution of a discrete random variable is the list of all possible values of the variable and their probabilities which sum to 1.

• The cumulative probability distribution function gives the probability that the random variable is less than or equal to a particular value.

• On the other hand, a continuous random variable can assume a continuum of values.

• The probability density function (PDF) (f(y)) is a non-negative continuous function that the area under f(y) between any points a and b is the probability that Y assumes a value between a and b.

• The total area under f(y) must be always 1.

• The probably most important probability distribution considered is the normal distribution.

• Normal distributions are symmetric and bell-shaped. A normal distribution is characterized by its mean \(\mu\) and its standard deviation \(\sigma\) - \(N(\mu,\sigma)\).

• The normal distribution has the PDF:

\[ f(x) = \frac{1}{\sqrt(2\pi\sigma)}exp-(x-\mu)^{2}/(2\sigma^{2}) \] - For the standard normal distribution we have \(\mu=0\) and \(\sigma =1\). Standard normal variates are often denoted by Z.

Learn more about pnorm(), dnorm(), qnorm(), rnorm() website

• An example of sampling with replacement is rolling a dice three times in a row.

# set seed for reproducibility
set.seed(1) # be able to get the same result - can be any number
# rolling a dice three times in a row
sample(1:6, 3, replace = T)

## [1] 1 4 1

# getting the mean of the sample
mean(sample(1:6,3, replace = T)) 

## [1] 3.33

• As we increase the number of times we role the dice, its mean converge to the its expected value - try!!!

• Other moment measures are the variance and the standard deviation. Both are measures of the dispersion of a random variable around its mean.

\[ \sigma^{2}_{Y} = Var(Y) = E[(Y-\mu_{Y})^{2}] = \sum_{i=1}^{k}(y_{i}-\mu_{y})^{2}p_{i} \] The standard deviation of Y is \(\sigma_{Y}\), the square root the variance. The units of the standard deviation are the same as the units of Y.

• The skewness of y is its expected cubed deviation from its mean:

\[ S = \frac{E(y-\mu)^{3}}{\sigma^{3}} \]

• It measures the amount of asymmetry in a distribution. The larger the absolute size of it, the more asymmetrical is the distribution.

• A large positive value indicate a long right tail, and large negative indicate a long left tail.

• A zero value indicates symmetry around the mean (normal)

• The kurtosis of y is the expected fourth power (moment) of the deviation of y from its mean

\[ K= \frac{E(y-\mu)^{4}}{\sigma^{4}} \] - It measures the thickness of the tails of a distribution.

• K>3 indicates ``fat tails’’ or leptokurtosis, relative to normal.

• Extreme events are more likely to occur than would be the case under normality.

• So far we have reviewed aspects of a known population distribution of random variables.

• However, most of time we have a sample of data drawn from an unknown population distribution, and we want to learn from the sample

\[ \{y_{t}\}_{t=1}^{T} \sim f(y) \]

• To do so, we use various estimators

• We can obtain estimators by replacing population expectations with sample averages

• The sample average is simply the arithmetic average

\[ \bar{y} = \frac{1}{T}\sum_{t=1}^{T}y_{t} \]

• It provides an empirical measure of the location of y

• The sample variance is the average squared deviation from the sample mean:

\[ \sigma^{2} = \frac{\sum_{t=1}^{T}(y_{t}-\bar{y})^2}{T} \]

• Sample variance:

\[ \hat{\sigma}^{2} = \frac{\sum_{t=1}^{T}(y_{t}-\bar{y})^2}{T} \] - we usually use a different version of \(\hat{\sigma}^{2}\), which corrects for the 1 degree of freedom used in the estimation of \(\bar{y}\), to estimate an unbiased estimator of \(\hat{\sigma}^{2}\):

\[ S^{2} = \frac{\sum_{t=1}^{T}(y_{t}-\bar{y})^2}{T-1} \]

• The sample skewness is:

\[ \hat{S} = \frac{1}{N} \times \frac{\sum_{t=1}^{T}(y_{t}-\bar{y})^3}{\hat{\sigma}^{3}} \]

• It provides an empirical measure of the amount of asymmetry in the distribution of y.

• The sample kurtosis:

\[ \hat{K} = N \times \frac{\sum_{t=1}^{T}(y_{t}-\bar{y})^4}{\hat{\sigma}^{4}} \]

• It provides an empirical measure of fatness of the tails of the distribution of y relative to a normal distribution.

# This package have contain daily historical stock prices
library(tidyquant)

data1<-FANG
head(data1)

## # A tibble: 6 × 8
##   symbol date        open  high   low close  volume adjusted
##   <chr>  <date>     <dbl> <dbl> <dbl> <dbl>   <dbl>    <dbl>
## 1 FB     2013-01-02  27.4  28.2  27.4  28    6.98e7     28  
## 2 FB     2013-01-03  27.9  28.5  27.6  27.8  6.31e7     27.8
## 3 FB     2013-01-04  28.0  28.9  27.8  28.8  7.27e7     28.8
## 4 FB     2013-01-07  28.7  29.8  28.6  29.4  8.38e7     29.4
## 5 FB     2013-01-08  29.5  29.6  28.9  29.1  4.59e7     29.1
## 6 FB     2013-01-09  29.7  30.6  29.5  30.6  1.05e8     30.6

ggplot(fb, aes(x=adjusted)) + geom_histogram()