# Lecture 01 - Solutions

## Exercise A - (5 min)

1. Why does this code throw an error? Try to fix it.
x <- 3; x > 2 & < 9
1. Does (NA & TRUE) equal (NA | TRUE)? Explain.
2. Does (Inf - Inf) equal (Inf - 1)? Explain.
3. Run the following. What happens? (further reading)
y <- (1 - 0.8); z <- 0.2
y == z; y < z; all.equal(y, z); identical(y, z)
1. Why do I use double quotes here?
important_message <- "The harder you try, the more you'll learn."

### Solution

# Part 1
# Need to write x twice to get two complete statements
x <- 3
(x > 2) & (x < 9)
[1] TRUE
# Part 2
NA & TRUE # result unknown since AND is only TRUE if both are TRUE
[1] NA
NA | TRUE # since one condition is true, OR is true
[1] TRUE
# Part 3
Inf - Inf
[1] NaN
Inf - 1
[1] Inf
# Part 4
y <- (1 - 0.8)
z <- 0.2
y == z
[1] FALSE
y < z
[1] TRUE
all.equal(y, z)
[1] TRUE
identical(y, z)
[1] FALSE
# Part 5
# With single quote, apostrophe in "you'll" would cause problems.

## Exercise B - (1 minute)

Predict the result that you will obtain if you use typeof() to find the type of each of the following atomic vectors. Then check to see if you were right!

foo <- c('1', '2', '3')
bar <- c('TRUE', 'FALSE')

### Solution

# They're both character vectors
typeof(foo)
[1] "character"
typeof(bar)
[1] "character"

## Exercise C - (5 minutes)

y <- c('Keble', 'LMH', 'Univ', 'Merton')
1. Enter the command y[5]. What result do you get? Why?
2. I want to extract the second and fourth elements of y so I enter y[2,4]. What happens? Can you fix it? How?
3. Select 'Keble' and 'Univ' two different ways.
4. Below is a vector of sales in $over several months. Using [], length() and -, compute the monthly growth rates in % sales <- c(100, 120, 90, 110, 105, 130, 140, 135, 125, 145, 150, 160) ### Solution y <- c('Keble', 'LMH', 'Univ', 'Merton') # Part 1 # We get an NA since there is no 5th element y[5] [1] NA # Part 2 # Need to enclose 2,4 within c() y[c(2, 4)] [1] "LMH" "Merton" # Part 3 y[c(1, 3)] [1] "Keble" "Univ"  y[-c(2, 4)] [1] "Keble" "Univ"  # Part 4 100 * ((sales[-1] / sales[-length(sales)]) - 1)  [1] 20.000000 -25.000000 22.222222 -4.545455 23.809524 7.692308 [7] -3.571429 -7.407407 16.000000 3.448276 6.666667 ## Exercise D - (3 min) The probability mass function of a Binomial$$(n, p)$$ random variable is given by $\mathbb{P}(X=x) = \binom{n}{x} p^x (1 - p)^{n-x}$ Use vectorized mathematical operations and the choose() function to calculate the pmf of a Binomial$$(5, 0.3)$$ random variable in one fell swoop. ### Solution n <- 5 p <- 0.3 x <- 0:n pmf <- choose(n, x) * p^x * (1 - p)^(n - x) pmf [1] 0.16807 0.36015 0.30870 0.13230 0.02835 0.00243 # Check that our calculations agree with dbinom() all.equal(dbinom(x, n, p), pmf) [1] TRUE ## Exercise E - (5 min) 1. Replace all of the 999s in this vector with NAs x <- c(5, 10, 3, 7, 999, 2, 999, 17, 0) 1. In a deck of Italian playing cards, the face cards are fante (Knave), cavallo (Knight), and re (King). In the game Scopa, fante is worth 8, cavallo 9, and re 10. Convert cards to the appropriate numeric values. cards <- c('re', 'cavallo', 're', 'fante', 'cavallo', 'fante', 're') 1. This code throws an error. Coerce y to make it work. y <- c('1', '2', '3') sum(y) 1. What happens if you run as.logical(-2:2)? Can you figure out the coercion rule for numeric to logical? ### Solution # Part 1 x[x == 999] <- NA x [1] 5 10 3 7 NA 2 NA 17 0 # Part 2 # The slickest solution uses a lookup table: lookup <- c('fante' = 8, 'cavallo' = 9, 're' = 10) cards <- c('re', 'cavallo', 're', 'fante', 'cavallo', 'fante', 're') lookup[cards]  re cavallo re fante cavallo fante re 10 9 10 8 9 8 10  # Part 3 y <- c('1', '2', '3') sum(as.numeric(y)) [1] 6 # Part 4 # Every element becomes TRUE except for 0, which becomes FALSE as.logical(-2:2) [1] TRUE TRUE FALSE TRUE TRUE ## Exercise F - (10 min) 1. Call z_score(w) where w <- c(1, 2, NA). What happens? See ?mean(). 2. Test out this function. What happens? Now try adding return(z) at the bottom of the function body. Explain your results. bad_z_score <- function(x) { z <- (x - mean(x)) / sd(x) } 1. Write a function to compute skewness using sum(), length(), mean() and sd(). $\text{Skewness} \equiv \frac{1}{n} \sum_{i=1}^n\left( \frac{x_i - \bar{x}}{s}\right)^3.$ 2. Use sum(), length() and is.na() to write a function called my_var() that drops NAs and then computes the sample variance. 3. Write a function called summary_stats() that returns a named vector with two elements: the sample mean and standard deviation. ### Solution # Part 1 # Part 2 # The final statement in this function *stores* the result so it doesn't return # anything. Either drop the assignment or add return() # Part 3 skewness <- function(x) { mean(((x - mean(x)) / sd(x))^3) } # Part 4 my_var <- function(x) { x <- x[!is.na(x)] n <- length(x) sum((x - mean(x))^2) / (n - 1) } # Part 5 summary_stats <- function(x) { c('mean' = mean(x), 'sd' = sd(x)) } ## Exercise G - (8 min) 1. What happens if you run the following code? Why? x <- c(TRUE, TRUE) if(x) { print('hello world!') } 1. What happens if you run this code? Try to fix it. if(3 > 5) { print('3 is greater than 5') } else { print('3 is not greater than 5') } 1. Write a function called mycov() that calculates the sample covariance between x and y. Use an early return to print an error message when x and y have different lengths. 2. Consult ?trunc(). Then use trunc() to write a function called myround() that rounds x to the nearest integer. ### Solution # Part 1 # This code fails: the condition inside of if() must evaluate to # a *single* logical value, but this is a vector. # Part 2 # The problem is the line break before else. This runs: if(3 > 5) { print('3 is greater than 5') } else { print('3 is not greater than 5') } [1] "3 is not greater than 5" # Part 3 mycov <- function(x, y) { if(!identical(length(x), length(y))) { return('Error: x and y must have the same length') } (x - mean(x)) * (y - mean(y)) } # Part 4 myround <- function(x) { integer_part <- trunc(x) decimal_part <- x - integer_part if(decimal_part <= 0.5) { out <- integer_part } else { out <- integer_part + 1 } out } ## Exercise H - (8 min) 1. The Fibonacci Sequence is defined by $$F_1 = 1$$, $$F_2 = 1$$ and $$F_n = F_{n-1} + F_{n-2}$$ for $$n > 2$$. Write a function that uses a for() loop to compute first n Fibonacci numbers. 2. Come up with a way to generate the same output as f() without using a loop or if() ... else. f <- $$x) { for(j in 1:length(x)) { if(x[j] > 0) { x[j] <- x[j]^3 + x[j] } else { x[j] <- x[j]^2 - x[j] } } x } ### Solution # Part 1 fib <- function(n) { out <- vector(length = n) out[2] <- out[1] <- 1 for(i in 3:n) { out[i] <- out[i - 1] + out[i - 2] } out } fib(12)  [1] 1 1 2 3 5 8 13 21 34 55 89 144 # Part 2 g <- function(x) { (x > 0) * (x^3 + x) + (x <= 0) * (x^2 - x) } f(-2:2) [1] 6 2 0 2 10 g(-2:2) [1] 6 2 0 2 10 ## Exercise I - (8 min) 1. Create a \(5\times 5$$ matrix called A, each of whose rows contains the elements 1:5. Hint: see ?rep. 2. Display all elements of A except row 3 and column 2. 3. Form a matrix B by stacking the $$(4\times 4)$$ identity matrix on top of itself. 4. Display the seventh row of B. 5. Write a function that uses a for() loop to construct the $$(n\times n)$$ exchange matrix $$J_n$$. ### Solution # Part 1 A <- matrix(rep(1:5, times = 5), 5, 5, TRUE) # Part 2 A[-3, -2]  [,1] [,2] [,3] [,4] [1,] 1 3 4 5 [2,] 1 3 4 5 [3,] 1 3 4 5 [4,] 1 3 4 5 # Part 3 B <- rbind(diag(nrow = 4), diag(nrow = 4)) # Part 4 B[7, ] [1] 0 0 1 0 # Part 5 get_exchange <- function(n) { out <- matrix(0, n, n) for(i in 1:n) { out[i, n + 1 - i] <- 1 } out } ## Exercise J - (8 min) 1. Write a function to constructs the $$(n\times n)$$ exchange matrix $$J_n$$ without using a loop. 2. Compute the element-wise product of $$J_3$$ with itself, and the square of $$J_3$$, i.e. the ordinary matrix product $$J_3 J_3$$. 3. Let $$X$$ be a Bernoulli$$(0.2)$$ and $$Y$$ be a Binomial$$(2, 0.5)$$ RV. Construct a matrix p_XY that represents the joint pmf of $$X$$ and $$Y$$, under the assumption that $$X$$ and $$Y$$ are independent. Name the rows and columns. 4. Consult ?rowSums() and ?colSums(). Then extract the marginal pmfs of $$X$$ and $$Y$$ from the matrix p_XY. ### Solution # Part 1 get_exchange <- function(n) { out <- matrix(0, n, n) anti_diagonal <- cbind(1:n, n:1) out[anti_diagonal] <- 1 out } # An even slicker solution to part 1, suggested by a student: get_exchange2 <- function(n) { diag(1, n)[n:1, ] } # Part 2 J3 <- get_exchange(3) J3 * J3  [,1] [,2] [,3] [1,] 0 0 1 [2,] 0 1 0 [3,] 1 0 0 J3 %*% J3  [,1] [,2] [,3] [1,] 1 0 0 [2,] 0 1 0 [3,] 0 0 1 # Part 3 p_XY <- c(0.2, 0.8) %o% c(0.25, 0.5, 0.25) rownames(p_XY) <- c('x=0', 'x=1') colnames(p_XY) <- c('y=0', 'y=1', 'y=2') # Part 4 rowSums(p_XY) x=0 x=1 0.2 0.8  colSums(p_XY)  y=0 y=1 y=2 0.25 0.50 0.25  ## Exercise K - (7 min) 1. I used students$name == 'Xerxes' above. Why didn’t I instead use identical(students$name, 'Xerxes')? 2. Use the following code chunk to construct the employees data frame. Then display it. employees <- data.frame( name = c("Alice", "Bob", "Cathy", "David", "Eva", "Frank", "Grace", "Hank", "Ivy", "Jack"), age = c(25, 31, 28, 40, 35, 23, 30, 45, 33, 29), department = c("HR", "IT", "Finance", "IT", "HR", "Finance", "IT", "HR", "Finance", "IT"), salary = c(50000, 60000, 55000, 70000, 53000, 51000, 62000, 71000, 57000, 59000) ) 1. Display the age column of employees. 2. Display the sixth row of employees. 3. Display the employee record for Eva. 4. Display employee records for everyone in the IT department. 5. Repeat the preceding, restricted to people with a salary of at least 60,000. ### Solution # Part 1 students <- data.frame('name' = c('Xerxes', 'Xanthippe', 'Xanadu'), 'age' = c(19, 23, 21), 'grade' = c(65, 70, 68), 'favorite_color' = c('blue', 'red', 'orange')) # identical() returns a *scalar* but we need a vector students[identical(students$name, 'Xerxes'), ] 
[1] name           age            grade          favorite_color
<0 rows> (or 0-length row.names)
# Part 2
employees
    name age department salary
1  Alice  25         HR  50000
2    Bob  31         IT  60000
3  Cathy  28    Finance  55000
4  David  40         IT  70000
5    Eva  35         HR  53000
6  Frank  23    Finance  51000
7  Grace  30         IT  62000
8   Hank  45         HR  71000
9    Ivy  33    Finance  57000
10  Jack  29         IT  59000
# Part 3
employees$age  [1] 25 31 28 40 35 23 30 45 33 29 # Part 4 employees[6, ]  name age department salary 6 Frank 23 Finance 51000 # Part 5 employees[employees$name == 'Eva', ]
  name age department salary
5  Eva  35         HR  53000
# Part 6
is_IT <- employees$department == 'IT' employees[is_IT, ]  name age department salary 2 Bob 31 IT 60000 4 David 40 IT 70000 7 Grace 30 IT 62000 10 Jack 29 IT 59000 # Part 7 high_salary <- employees$salary >= 60000
employees[is_IT & high_salary, ]
   name age department salary
2   Bob  31         IT  60000
4 David  40         IT  70000
7 Grace  30         IT  62000