Individual Assignment 1
Question 1
a) A short e-survey is released, asking the following question: “How many cups of tea do you drink a day?”. The available answers were “0”, “1”, “2“, and “3 or more”.
What type of data will the answers to the e-survey give us?
i) Nominal
ii) Ordinal
iii) Discrete
iv) Continuous
b) A random variable U is defined with the probability density function below:
Which of the following is the value of P(U < 0.5)?
i) 0.2174
ii) 0.4375
iii) 0.7022
iv) 0.9775
Question 2
The dataset airquality in R gives information about the air quality in New York City in 1973. It is made up of six variables, but the two we will focus on are both daily measurements taken at LaGuardia Airport - the maximum daily temperature in degrees Farenheit (Temp), and the average wind speed in miles per hour at 7am and 10am (Wind).
a) Create a histogram of the Temp data, and include it in your submission. State whether this histogram shows that the Temp data is skewed, and if so, whether the skew is positive or negative.
b) Create a scatterplot comparing the Temp and Wind data, and include it in your submission. Comment on the relationship between the maximum temperature at LaGuardia in a day, and the average speed of the wind at 7am and 10am at LaGuardia on the same day. (What I am asking here is for you yo comment on any overall relationship between max temperature and average windspeed, that gives us some sense of how daily maximum temperature and daily average windspeed are related.)
Question 3
An understanding of probability can sometimes be used to detect cases of fraud and deception. Here is a (very small and minor) example of such a process.
I ask a student to help in an experiment. They are asked to perform. a task 200 times. Each task is the same - they must toss a fair coin six times, and write down the sequence of Heads (H) and Tails (T) results they get. Note that, as the coin is fair, P(H) = P(T) = 1/2 for each toss.
a) In this question, we will assume the result of each coin toss will be independent of the results of all other coin tosses. Explain, in language referencing the specific situation described in this question, what it means to assume independence in this way.
b) Denote by Xi the random variable which represents the number of H results for the ith task, i = 1, . . . , 200. State, giving the value of each parameter, the distribution each Xi must have.
c) Find the probability that all six results for a single task are H. NOTE: You will need to show your working.
d) Find the expected number of the 200 sequences which will show six H results.
I am aware that the student might not actually perform. the task, and instead just write out 200 sequences of six letters each to save time. If they do this, I know that they will try to make those sequences look like they’d come from the results of a fair coin.
Luckily, there is a way to test whether this may have happened. When someone cheats in this kind of experiment, they almost never write a sequence which is all H results, or all T results. It doesn’t look “random” enough for them. As the answer to 1.3d) shows, though, we expect such results happen at least occasionally.
e) Let Y be the random variable which expresses the number of the 200 sequences which contain six H results, assuming the student did correctly perform. the task set. State, giving the value of each parameter, the distribution Y must have.
f) Find the probability that Y = 0; that is, the probability that no sequence comprises of six heads, assuming the student correctly performed the task. NOTE: You will need to show your working.
g) The student hands in their 200 sequences, and there are indeed no sequences comprising of six H results. A friend tells you that your answer to 1.3f) gives the probability the student did not cheat. State, giving an explanation, whether your friend is correct or incorrect.
Question 4
Let X ~ Pois(0.7), and let Y ~ Pois(λ). Find:
1. P(X = 1).
2. P(X > 1). NOTE: You will need to show your working.
3. An algebraic expression for P(Y = 4|Y > 1) in its simplest form. NOTE: You will need to show your working.