## Introduction

We will learn how to compute basic statistics in R by using the subset of beef cattle data set.

We will first set a path to the phenotypic dataset. Insert your path to the file inside the double quotation " ". Your path may differ from the path below if you stored the file in a different directory.

FILE <- "../data/sub/1000withEffects.redangus"

The function read.table reads a file in a data frame format. Typing help will open a documentation page.

The read.table function has many arguments but we will use only three in this exercise.

The head function returns the first six rows of a data frame. The number of rows returned can be varied by controlling the argument n.

The dim function returns the dimension of data frame.

dim(dat)

We will use two phenotypes to compute basic statistics. Each column of data frame can be accessed by the $operator followed by a column name. dat$BWT
dat$CE The length function returns the length of vector. length(dat$BWT)
length(dat$CE) ## Mean The mean function computes the mean of a vector. mean(dat$BWT)

### Exercise 1

Verfiy that $$E(aX) = aE(X)$$. Set $$a = 10$$. Here $$X$$ is a vector of body weight. Use the multiplication operator *.

## Variance

The var function computes the sample variance of a vector.

var(dat$BWT) Alternatively, we can use the equation $$Var(X) = \frac{1}{N-1}\sum(X - \bar{X})^2$$ sum((dat$BWT - mean(dat$BWT))^2)/(length(dat$BWT) - 1)

### Exercise 2

Verfiy that $$Var(aX) = a^2Var(X)$$. Set $$a = 10$$.

## Covariance

The cov function computes the covariance of two vectors.

cov(dat$BWT, dat$CE)

Alternatively, we can use the equation $$Cov(X, Y) = \frac{1}{N-1}\sum(X - \bar{X})(Y - \bar{Y})$$

### Exercise 5

Verfiy that regression of $$Y$$ on $$aX$$ is $$b_{Y, aX} = \frac{Cov(aX,Y)}{Var(aX)} = \frac{Cov(X,Y)}{aVar(X)}$$. Set $$a = 10$$.