Name:
Andrew ID:
Collaborated with:
This lab is to be done in class (completed outside of class time if need be). You can collaborate with your classmates, but you must identify their names above, and you must submit your own lab as an knitted PDF file on Gradescope, by Friday 9pm, this week.
## For reproducibility --- don't change this!
set.seed(08312021)This week’s agenda: manipulating data objects; using built-in functions, doing numerical calculations, and basic plots; reinforcing core probabilistic ideas.
The binomial distribution \(\mathrm{Bin}(m,p)\) is defined by the number of successes in \(m\) independent trials, each have probability \(p\) of success. Think of flipping a coin \(m\) times, where the coin is weighted to have probability \(p\) of landing on heads.
The R function rbinom() generates random variables with a binomial distribution. E.g.,
rbinom(n=20, size=10, prob=0.5)produces 20 observations from \(\mathrm{Bin}(10,0.5)\).
bin.draws.0.5. Extract and display the first 25 elements. Extract and display all but the first 475 elements.# YOUR CODE GOES HEREbin.draws.0.5 to the fifth. Compare the second element to the tenth, which is larger? A bit more tricky: print the indices of the elements of bin.draws.0.5 that are equal to 3. How many such elements are there? Theoretically, how many such elements would you expect there to be? Hint: it would be helpful to look at the help file for the rbinom() function.# YOUR CODE GOES HEREbin.draws.0.5. Is the mean close what you’d expect? The standard deviation?# YOUR CODE GOES HEREsummary() on bin.draws.0.5 and describe the result.# YOUR CODE GOES HEREbin.draws.0.5 using typeof(). Then convert bin.draws.0.5 to a vector of characters, storing the result as bin.draws.0.5.char, and use typeof() again to verify that you’ve done the conversion correctly. Call summary() on bin.draws.0.5.char. Is the result formatted differently from what you saw above? Why?# YOUR CODE GOES HEREplot() is a generic function in R for the visual display of data. The function hist() specifically produces a histogram display. Use hist() to produce a histogram of your random draws from the binomial distribution, stored in bin.draws.0.5.# YOUR CODE GOES HEREtabulate() on bin.draws.0.5. What is being shown? Does it roughly match the histogram you produced in the last question?# YOUR CODE GOES HEREplot() on bin.draws.0.5 to display your random values from the binomial distribution. Can you interpret what the plot() function is doing here?# YOUR CODE GOES HEREplot() with two arguments, the first being 1:500, and the second being bin.draws.0.5. This creates a scatterplot of bin.draws.0.5 (on the y-axis) versus the indices 1 through 500 (on the x-axis). Does this match your plot from the last question?# YOUR CODE GOES HEREbin.draws.0.2, bin.draws.0.3, bin.draws.0.4., bin.draws.0.6, bin.draws.0.7 and bin.draws.0.8. For each, compute the mean and standard deviation.# YOUR CODE GOES HERE# YOUR CODE GOES HERE# YOUR CODE GOES HEREbin.matrix, whose columns contain the 7 vectors we’ve created, in order of the success probabilities of their underlying binomial distributions (0.2 through 0.8). Hint: use cbind().# YOUR CODE GOES HEREbin.matrix. Print the element in the 66th row and 5th column. Compute the largest element in first column. Compute the largest element in all but the first column.# YOUR CODE GOES HEREbin.matrix by using just a single function call.# YOUR CODE GOES HERE==, and second, using identical(). What do the two ways report? Are the results compatible? Explain.# YOUR CODE GOES HEREbin.matrix and then take row means. Are these the same as what you just computed? Should they be?# YOUR CODE GOES HEREbig.bin.draws. Calculate the mean and standard deviation of this vector.# YOUR CODE GOES HEREbig.bin.draws.standardized, which is given by taking big.bin.draws, subtracting off its mean, and then dividing by its standard deviation. Calculate the mean and standard deviation of big.bin.draws.standardized. (These should be 0 and 1, respectively, or very close to it; if not, you’ve made a mistake somewhere).# YOUR CODE GOES HEREbig.bin.draws.standardized exceeds 1.644854. Is this close to 0.05?# YOUR CODE GOES HEREhuge.bin.draws.# YOUR CODE GOES HEREhuge.bin.draws. Are they close to what you’d expect? (They should be very close.) Did it longer to compute these, or to generate huge.bin.draws in the first place?# YOUR CODE GOES HEREhuge.bin.draws. Did this median calculation take longer than the calculating the mean? Is this surprising?# YOUR CODE GOES HEREhuge.bin.draws, in one line of code. Did this take longer than the median calculation applied to huge.bin.draws directly? Is this surprising?# YOUR CODE GOES HEREhuge.bin.draws, again with a large setting of the breaks argument (e.g., breaks=100). Describe what you see; is this different from before, when we had 3 million draws? Challenge: Is this surprising? What distribution is this?# YOUR CODE GOES HEREbig.bin.draws into a list using as.list() and save the result as big.bin.draws.list. Check that you indeed have a list by calling class() on the result. Check also that your list has the right length, and that its 1159th element is equal to that of big.bin.draws.# YOUR CODE GOES HEREbig.bin.draws.list. Note that lapply() applies the function supplied in the second argument to every element of the list supplied in the first argument, and then returns a list of the function outputs. (We’ll learn much more about the apply() family of functions later in the course.) Did this lapply() command take longer to evaluate than the code you wrote in Q5b? (It should have; otherwise your previous code could have been improved, so go back and improve it.) Why do you think this is the case?big.bin.draws.mean = mean(big.bin.draws)
big.bin.draws.sd = sd(big.bin.draws)
standardize = function(x) {
return((x - big.bin.draws.mean) / big.bin.draws.sd)
}
big.bin.draws.list.standardized.slow = lapply(big.bin.draws.list, standardize)big.bin.draws.list, using lapply(). Why is it so much slower than the code in the last question? (You may stop evaluation if it is taking too long!) Think about what is happening each time the function is called.standardize.slow = function(x) {
return((x - mean(big.bin.draws)) / sd(big.bin.draws))
}
big.bin.draws.list.standardized.slow = lapply(big.bin.draws.list, standardize.slow)object.size(x) returns the number of bytes used to store the object x in your current R session. Find the number of bytes used to store big.bin.draws and big.bin.draws.list. How many megabytes (MB) is this, for each object? Which object requires more memory, and why do you think this is the case? Remind yourself: why are lists special compared to vectors, and is this property important for the current purpose (storing the binomial draws)?# YOUR CODE GOES HERE