---
title: "STAT 2450 Assignment 1, Winter 2020 - due January 20, at beginning of class"
author: "Your name here"
date: 'Banner:  B00??????'
output:
  html_document: default
  pdf_document: default
  word_document: default
---
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1.  there are approximately 8 kilometres in 5 miles.  There are approximately 4.55 litres in one imperial gallon.
    An automobile uses 9.5 litres of gasoline while traveling 100 kilometers.  What is the mileage in miles per gallon?

```{r}
Nk=100 # number of kilometres travelled
Lg=9.5 # number of litres of gasoline used
Nm=100*5/8 #number of miles travelled
Gg= NA #number of gallons used. currently set to missing. replace with appropriate formula.
MPG=NA #replace with appropriate formula
MPG
```

2.  The probability that an exponential random variable $T$ with mean $\theta$ is greater than $t_0$ is given by

 $$P(T>t_0) = e^{-t_0/\theta}$$


   Calculate the probability that an exponential random variable with mean 10 is greater than 20.

```{r}
theta=10
t0=20
prob=NA #enter the appropriate formula here
prob
```

3.  The probability density function of a $\chi^2$ random variable with $k$ degrees of freedom, evaluated at
the point $x$, is given by
$$f(x) = \frac{x^{k/2-1} e^{-x/2} }{2^{(k/2)} ((k/2)-1)!}$$

where $(k-1)!$ is "k-1 factorial", which in R is given by "factorial(k-1)"

Use R to calculate the probability density function of a $\chi^2$ random variable with 6 degrees of freedom, evaluated
at the point $x=4$.

```{r}
x=4
k=6
fx=NA #write the R expression for $f(x)$ here, using the constant "e", the "factorial" function, and other
      #arithmetic operations
fx 
```

Note:  you can check your result using the builtin R function "dchisq", using the syntax "dchisq(4,6)".


4.  The file http://mathstat.dal.ca/~fullsack/stat2450/Data/A14.txt contains a
sequence of 12 numbers.

    + Read the numbers into a vector using the "scan" command. Print the vector.
    + Arrange the 12 numbers into a matrix with 3 rows and 4 columns, where the first 4 numbers in the vector are in the first row, the next 4 numbers in the second row, and so on.  Use the matrix command, with the "byrow=T" argument. Print the matrix.
    +  Arrange the 12 numbers into a 4 by 3 matrix, where the first 3 numbers in the vector are in the first row, the next 3 numbers in the second row.  Print the matrix.
    + Calculate, and print,  the matrix product of the the first and second matrices, using the matrix multiplication operator "%*%. 
    + Use the "det" command to calculate and print the determinant of the product matrix
    + Use the "dim" command to get the dimensions of the product matrix. 
    +  extract the second row of the product matrix into a vector V2, and print

```{r}
#V=scan("put in the appropriate pathname here")
#V
#M1=matrix(V,byrow=T,nrow=3)
#M1
#M2= enter appropriate syntax here.
#M2
#Mprod = enter appropriate syntax here
#Mprod
#det(Mprod)  calculate the determinant
#Mdim= enter appropriate syntax here
#Mdim
#V2= enter appropriate syntax here
```

5. The file 

    http://faculty.marshall.usc.edu/gareth-james/ISL/Auto.csv
    contains data on an outcome variable "mpg", which is automobile mileage
in miles per gallon, and a number of other variables which
can be used to predict mileage.

    + Use the **read.csv** command to read the data into a data frame.
The syntax for this is in the notes "myR_Basics1".  
    + Use the summary
command to get summary statistics for all of the variables in the data
set.  For example, if you read the data into a dataframe named
"mydata", then you would type summary(mydata) to get the
    + Make boxplots of the mileage values for the different numbers of
cylinders. Make sure to add appropriate labels to the X and Y axes,
and add a title to the plot. The syntax for the boxplot command is in
the Rintro notes.  In order to do this, you need to either attach the
created data frame prior to using the plot command, or include the
"data=mydata" option on the boxplot command 
    + Make a scatterplot with miles per gallon on
the Y axis and displacement on the X axis. Once again, make sure to
label the axes and add a title.  If you have "attached" the data
frame, you won't need the "data=mydata" syntax on the command line.
Otherwise, you will.

```{r}
#yourdata=read.csv(  enter appropriate syntax here
#summary(mydata)
#boxplot(appropriate syntax here, maybe need to include data=mydata)
#plot(appropriate syntax here,  maybe need to include data=mydata)
```
The displacement values are in cubic inches, where 1 cubic inch equals
`r 2.54^2` cubic centimetres, so a 400 cubic inch engine had a displacement of `r 400*2.54^3`.  The mileage was correspondingly dismal.  You could refer back to question 1 to see what 10 mpg, pretty common back in the day, corresponds to in litres per 100 kilometres.
Conservation of resources wasn't high on the typical consumer's prioity list. 

