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Today is Day 14

Between Class Sessions - Prep for Day 14

Please spend around 2 hours working between class sessions, focusing on the tasks below. Use any extra time to complete KnewtonAlta assignments and/or work on Project tasks.

Pick something from your prep today that you can share with your group in class. It might be something new that you learned. It might be questions you have that are still unanswered, or a question along with what helped you eventually answer it. It might be something tricky that you solved. It might be a review topic that helped you remember something. It might be a conversation you had with AI that was helpful. You will have a chance to share this with your peers during class. Come ready to articulate your thinking and questions.

Preparation

(1) Probability Exercise (Tropical Storms Example)

Read the definitions and complete the exercises below.

prod(1:3)
factorial(3)

The number of tropical storms in Florida in a given year can be thought of as a Poisson random variable, which means we can model the probabilty of having $x$ tropical storms in a year using $$p(x; \lambda) = \frac{\lambda^x}{x!}e^{-\lambda}\text{ with }x = 0, 1, 2, 3, ...\text{ where }\lambda > 0.$$ The parameter $\lambda$ is a value we would like to determine. The code chunk below defines this probability mass function in R, using a default value of $\lambda = 2$. Start by exploring the function $p(x; \lambda)$ and how the parameter $\lambda$ affects the plot.

p <- function(x,lambda=2){
# x must be a whole number
  (lambda^x/factorial(x))*exp(-lambda)
}

x <- seq(0,10,1) # x must be a whole number
plot(x,p(x))
plot(x,p(x,lambda=5))
plot(x,p(x,lambda=2.4)) # lambda just needs to be positive.

Assuming the parameter $\lambda = 2$, calculate the following and check your answers.

• The probability of 4 Florida tropical storms this year.
• The probability of 5 Florida tropical storms this year.
• The probability of 2 Florida tropical storms this year.
• The probability of 3 or more Florida tropical storms this year.
• The probability of more than 10 Florida tropical storms this year.
• The probability of 2 or less Florida tropical storms this year.

x <- 0:30

par(mfrow=c(1,1),mar=c(2.5,2.5,0.25,0.25))
barplot(p(x),ylim=c(0,0.3),width=rep(1,length(x)),space=1)

p(4)
p(5)
p(2)
1 - (p(0) + p(1) + p(2))
1 - sum(p(0:10))
sum(p(0:2))
    1-(1 - (p(0) + p(1) + p(2)))

Now answer the following and come with answers to compare with your group in class.

• The probability of less than 8 Florida tropical storms this year using $\lambda = 3$.
• The probability of less than 8 Florida tropical storms this year using $\lambda = 6$.
• The probability of at least 8 Florida tropical storms this year using $\lambda = 6$.
• The probability of more than 12 Florida tropical storms this year using $\lambda = 5$.

Regular Reminders

Skill Practice (KA Homework)

• Complete 3 -- Probability Basics assignment in Knewton Alta

Applied Practice (Project Work)

• Complete and Submit Project 1 Task 3

During Class

Brain Gains

Let $ h(t) = \sqrt{3t+1} $.

• Compute $ h(2) $.
• Solve $ h(t) = 2 $ for $ t $.

Let $X_1$, $X_2$, and $X_3$ be independent random variables which each return the value of a single toss of one fair six-sided die.

• Compute $P(X_1 = 1, X_2 = 4, \text{ and } X_3 = 5)$.
• Compute $P(X_1 \leq 3, X_2\leq 3,\text{ and }X_3\leq 3)$.

Recall that the probability mass function (pmf) for a Poisson distribution is $p(x; \lambda) = \frac{\lambda^x}{x!}e^{-\lambda}$ with $x = 0, 1, 2, 3, ...$ where $\lambda > 0$. Suppose that $Y$ is a random variable with a Poisson distribution with $\lambda = 1$.

• Compute $P(Y=1)$.
• Compute $P(Y=0)$.
• Compute $P(Y>1)$.

You may use the code below to help you code in the Poisson distribution.

p <- function(x,lambda=2){
# x must be a whole number
  (lambda^x/factorial(x))*exp(-lambda)
}

p(1,1)
p(0,1)
1-(p(0,1)+p(1,1))

Group Meeting

Start by giving each person a moment to share what they chose to prepare for class. Help each other address any questions. When each person has had a chance to share, move on the other activities.

Probability Exercise (Tropical Storms Example from Prep)

Between class we worked with a Poisson Random Variable to model the number of tropical storms in Florida in a given year. The probability mass function for a Poisson distribution is given by $p(x; \lambda) = \frac{\lambda^x}{x!}e^{-\lambda}$ with $x = 0, 1, 2, 3, ...$ where $\lambda > 0$. You can use the following code to use this function in R.

p <- function(x,lambda=2){
# x must be a whole number
  (lambda^x/factorial(x))*exp(-lambda)
}

Compare your answers that you got for the last 4 questions of yesterday's prep work. These 4 problems asked you to compute each of the things below.

• The probability of less than 8 Florida tropical storms this year using $\lambda = 3$.
• The probability of less than 8 Florida tropical storms this year using $\lambda = 6$.
• The probability of at least 8 Florida tropical storms this year using $\lambda = 6$.
• The probability of more than 12 Florida tropical storms this year using $\lambda = 5$.

Activity - Florida Tropical Storms Revisited

Read through the following work, execute the code chunks, and then complete the additional calculations and discuss the questions shown at the end.

Let $X$ be the random variable that counts the number of tropical storms in Florida this year. Assuming the parameter $\lambda = 2$, we calculate the following:

#Define the Poisson distribution with a default value for lambda
p <- function(x,lambda=2){
# x must be a whole number
  (lambda^x/factorial(x))*exp(-lambda)
}

#The probability of 4 Florida tropical storms this year (lambda = 2 is assumed)
p(4) 

#The probability of 4 Florida tropical storms this year (lambda = 2 is assigned - should match above. )
p(4,2) 

#The probability of 5 Florida tropical storms this year (lambda = 2 is assumed)
p(5)

#The probability of 2 Florida tropical storms this year (lambda = 2 is assumed)
p(2)

#The probability of $x$ Florida tropical storms this year (lambda = 2 is assumed) for each x from 0 to 10
p(0:10)

The value for $\lambda$ does not have to be 2. We can instead change it to 5, or some other value, and repeat the computations above.

#The probability of 4 Florida tropical storms this year, using lambda = 5. 
p(4,5) 

#The probability of $x$ Florida tropical storms this year, using lambda = 5, for each x from 0 to 10
p(0:10,5)

#The probability of $x$ Florida tropical storms this year, using lambda = 1, for each x from 0 to 10
p(0:10,1)

Additional Calculations and Questions

• Compute the probability of 7 Florida tropical storms in a year assuming $\lambda = 8$.
• Compute the probability of 7 Florida tropical storms in a year assuming $\lambda = 4$.
• If we actually observed of 7 Florida tropical storms in a year, would you be more likely to say that $\lambda =8$ or $\lambda = 4$? Why?
• Compute the probability of 7 Florida tropical storms in a year assuming $\lambda = 6.3$. (Wait, can we use decimals for $\lambda$?)
• Compute the probability of 7 Florida tropical storms in a year using various values for $\lambda$. Then as a group decide what value you think is the best value to assume for $\lambda$ if we actually did see 7 tropical storms in a year.
• Construct a plot that has $\lambda$ on the horizontal($x$) axis, and on the vertical($y$) axis we place the probability of 7 Florida tropical storms in a year assuming that value for $\lambda$. Describe the shape of this plot, and how can we use it to find the "best value" for $\lambda$.
• How does your choice of "best" $\lambda$ change if we know there will be 4 tropical storms (rather than 7 storms)?
• How does your choice for $\lambda$ change if we know there will be 5 tropical storms (rather than 7 storms)?

Activity - Florida Tropical Storms Extended

Let $X_1$ be the random variable that counts the number of tropical storms in Florida this year, $X_2$ be the random variable that counts the number of tropical storms in Florida next year, and $X_3$ be the random variable that counts the number of tropical storms in Florida the year after. Assuming $X_1$, $X_2$, and $X_3$ are independent Poisson random variables each with the the same parameter $\lambda = 2$. This means we have $$p_{3yr}(x_1, x_2, x_3; \lambda) = \left(\frac{\lambda^{x_1}}{x_1!}e^{-\lambda}\right) \left(\frac{\lambda^{x_2}}{x_2!}e^{-\lambda}\right) \left(\frac{\lambda^{x_3}}{x_3!}e^{-\lambda}\right) = \frac{\lambda^{x_1 + x_2 + x_3}}{x_1!x_2!x_3!}e^{-3\lambda}$$ with $x_1 = 0, 1, 2, 3, ...$, $x_2 = 0, 1, 2, 3, ...$, and $x_3 = 0, 1, 2, 3, ...$, (and zero otherwise) where $\lambda > 0.$ The following code chunk provides two versions to input this function. The first requires you input a vector $x$ consisting of the number of tropical storms that occurred each year (and can be used if you want to look at a period of 3 years, or even 7 years, by just changing the number of inputs to the vector). The second version will only work with a 3 year period.

p3v1 <- function(x,lambda=2){
# each element of x must be a whole number
  prod((lambda^x/factorial(x))*exp(-lambda))
}

p3v2 <- function(x1,x2,x3,lambda=2){
# x1, x2, and x3 must be whole numbers
  (lambda^(x1+x2+x3)/(factorial(x1)*factorial(x2)*factorial(x3)))*exp(-3*lambda)
}

We can now calculate the following probabilities:

#The probability of 4 Florida tropical storms this year, 4 Florida tropical storms next year, and 8 Florida tropical storms the year after (using $\lambda = 2$ as assumed).
p3v1(c(4,4,8))
#The same probability as above, using the other version
p3v2(4,4,8)
#The same probabilty as above by just multiplying probabilities of independent events together. 
p(4)*p(4)*p(8)

#The probability of 2 Florida tropical storms this year, 5 Florida tropical storms next year, and 3 Florida tropical storms the year after.
p3v1(c(2,5,3))

Again, we can change $\lambda$ to another value, and see how these probabilities change.

#The probability of 4 Florida tropical storms this year, 4 Florida tropical storms next year, and 8 Florida tropical storms the year after, assuming the parameter $\lambda = 5$.
p3v1(c(4,4,8),5)

#Repeat the above, but with lambda = 1, and then with lambda = 10
p3v1(c(4,4,8),1)
p3v1(c(4,4,8),10)

p3v2(4,4,8,1)
p3v2(4,4,8,10)

If we know there will be 4 Florida tropical storms this year, 4 Florida tropical storms next year, and 8 Florida tropical storms the year after, then our calculations above can help us determine a good value for the parameter $\lambda$. The value $\lambda=10$ is a better option that $\lambda = 1$ because $p3(4,4,8,10) = 4.029234e-05 > 2.143747e-09 = p3(4,4,8,1)$.

• Try various other values for $\lambda$ and calculate the probability of 4 Florida tropical storms this year, 4 Florida tropical storms next year, and 8 Florida tropical storms the year after.
• What criteria would you use to decide "what is the best value for the parameter $\lambda$?"
• Construct a plot (use p3v2) that has $\lambda$ on the horizontal($x$) axis, and on the vertical($y$) axis we place the probability of 4, then 4, then 8 Florida tropical storms for the next three years assuming that value for $\lambda$. Describe the shape of this plot, and how can we use it to find the "best value" for $\lambda$.
• In the plot you made, update your code to use p3v1 instead of p3v2. If you notice an error, that's OK. The problem is that R does not know how to use two vectors as an input to $p$ without more information. The sapply function deals with this and shows up in the prep tomorrow.
• Use the data from https://en.wikipedia.org/wiki/List_of_Florida_hurricanes_(200080%93present) for the years 2000-2005 (6 years) to estimate a good value for lambda (you'll need to use p3v1 now, as there are 6 data points instead of 3).

Between Class Sessions - Prep for Day 15

Please spend around 2 hours working between class sessions, focusing on the tasks below. Use any extra time to complete KnewtonAlta assignments and/or work on Project tasks.

Pick something from your prep today that you can share with your group in class. It might be something new that you learned. It might be questions you have that are still unanswered, or a question along with what helped you eventually answer it. It might be something tricky that you solved. It might be a review topic that helped you remember something. It might be a conversation you had with AI that was helpful. You will have a chance to share this with your peers during class. Come ready to articulate your thinking and questions.

Preparation

(1) Likelihood Practice (with Poisson)

Use the storms listed on Wikipedia's Florida Tropical Storms page to answer the questions below.

1. Use the data from 2005-2007 for $x_1$, $x_2$, and $x_3$. The number of storms in 2005 is $x_1$, the number of storms in 2006 is $x_2$, and the number of storms in 2007 is $x_3$.

• Adapt the code from class to complete the following table, calculating the probability given the value of $\lambda$.

p3v1 <- function(x,lambda=2){
# each element of x must be a whole number
  prod((lambda^x/factorial(x))*exp(-lambda))
}

p3v1(c(8,2,8),1)
p3v1(c(8,2,8),2)

• Use the code below to plot this function of $\lambda$ in R. (Because both $\lambda$ and $x$ are vectors, we must use the sapply function from R to appropriately tell the computer how to to parse the notation. You do not have to modify any code involving sapply for future tasks, so feel free to just run the code below to produce the needed plot.)

lambda <- seq(0,15,0.01)
x <- c(8,2,8)
probs <- sapply(lambda, p3v1, x = x)
plot(lambda, probs, type='l')

• If we know there will be 8 Florida tropical storms this year, 2 Florida tropical storms next year, and 8 Florida tropical storms the year after that (in other words the data values from 2005-2007), what is your guess for the best choice of parameter $\lambda$?

2. Using the data from 2010-2012 adapt the code from exercise 1 (above) to plot a function with $\lambda$ on the horizontal axis and probability (that of the number of storms this year will be the same as the number of storms in 2010, the number of storms next year will be the same as the number of storms in 2011, and the number of storms in the year after that will be the same number as the number of storms in 2012) on the vertical axis. See Wikipedia's Florida Tropical Storms to get the appropriate data.

• Using your plot what is your guess for the best choice of the parameter $\lambda$ in this case?

3. Using the data from 2012-2015 adapt the code to plot a function with $\lambda$ on the horizontal axis and probability (that of the number of storms this year will be the same as the number of storms in 2012, the number of storms next year will be the same as the number of storms in 2013, the number of storms in the year after that will be the same number as the number of storms in 2014, and the number of storms in the year after that will be the same number as the number of storms in 2015) on the vertical axis.

• Note: There are 4 years this time. How will you have to change the code to allow for this?
• Using your plot what is your guess for the best choice of the parameter $\lambda$ in this case?

4. Using the data from 2000 and 2001 adapt the code to plot a function with $\lambda$ on the horizontal axis and probability on the vertical axis.

• Note: There are 2 years this time. How will you have to change the code to allow for this?
• Using your plot what is your guess for the best choice of the parameter $\lambda$ in this case?

5. Using the data from 2019 and 2020 adapt the code to plot a function with $\lambda$ on the horizontal axis and probability on the vertical axis.

• Note: There are 2 years this time. How will you have to change the code to allow for this?
• Using your plot what is your guess for the best choice of the parameter $\lambda$ in this case?

(2) Residuals Exercises

Consider the 6 functions you have been fitting in Project 1.

• Write down a formula for the residual given model $f_1$.
• Write down a formula for the residual given model $f_2$.
• Write down a formula for the residual given model $f_3$.

Regular Reminders

Applied Practice (Project Work)

• Complete Project 1 Task 4

Today

• Jan26, Feb, Mar, Apr