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).