Class 35

Between Class Sessions - Prep for Day 35

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) Reading - Rectangles, Targets, and Sums - Sections 2.3 through 3.2

For this Unit, the reading as well as many practice exercises appear in the document Rectangles, Targets, and Sums

  1. Read sections 2.3-3.2.

  2. Complete a few exercises from each section, and come ready to share what you did with your group.

  3. Let \(g(x) = 2x\) for \(0\leq x\leq 4\) be a target function.

    • Find a value \(k\) so that the area under \(f(x) = k g(x)\) is one (so provide a normalized target function).
    • Let \(X\) be the random variable which records the \(x\)-coordinate of dropping a dart on the target defined by \(g\). Find the cumulative distribution function \(F(x)\).
    • Compute the derivative \(F'(x)\)
Solutions for 3

  • The area under the function \(g\) is \(A = \frac{1}{2}(4)(8) = 16\), which means we should pick \(k = \frac{1}{\text{Area}} = \frac{1}{16}\) so that the area under the function \(f\) is 1. This means \(f(x) = kg(x) = \frac{1}{16}(2x) = \frac{1}{8}x\) for \(0 \leq x \leq 4\).

  • The cdf of \(X\) will be the following function. \(F(x) = \begin{cases} 0 & \quad x \leq 0 \\ \\ \frac{x^2}{16} & \quad 0 < x < 4 \\ \\ 1 & \quad x \geq 4. \end{cases}\)

  • Calculating the derivative of \(F(x)\) we find. \(F'(x) = \begin{cases} 0 & \quad x < 0 \\ \\ \frac{1}{16}(2x) & \quad 0 < x < 4 \\ \\ 0 & \quad x > 4 \end{cases} = \begin{cases} \frac{1}{8}(x) & \quad 0 < x < 4 \\ \\ 0 & \quad \text{otherwise} \\ \\ \end{cases}\)

Regular Reminders

Skill Practice (KA Homework)

If you have not done so already, complete 3 – Discrete Random Variables and Expected Value.

Applied Practice (Project Work)

  • Continue working on Project 3 Task 1
    • Visually fit the probability models \(f_2\) and \(f_3\) to the data.
    • Use the parameters for your visually fitted models to simulate data and calculate approximate probability.

During Class

Brain Gains

We’re learning new vocabulary in this unit. What we need is some repeated practice to help us cement what that vocabulary means.

  1. What is the expected value for the random variable obtained by recording the \(x\)-coordinate of a dart thrown at a rectangular target?

  2. Let \(X\) represent the random variable obtained by recording the \(x\)-coordinate of a dart that is randomly thrown at a target that lies under the function \(\ds g(x) = \begin{cases}4&0\leq x \leq 2\\1&2 < x\leq 5 \\ 0 &\text{otherwise} \end{cases}\).

    1. Draw the function.

  3. Find the area under \(g\).

  4. Give the normalized target function by finding the constant \(k\) so that the area under \(f(x) = kg(x)\) is 1.

  5. Compute \(P(X\leq 3)\).

  6. Calculate the expected value of \(X\), so compute \(E[X]\).

  7. Give a formula for \(F(x)\), the cumulative distribution function.

  8. Compute \(F'(x)\).

Solutions
  1. I’ll let you graph it.

  1. The area is \(A = 11\).

  2. The value \(k\) is always \(k =\frac{1}{A}\), which gives \(k = \frac{1}{11}\) and so \[\ds f(x) = \begin{cases}\frac{4}{11}&0\leq x\leq 2\\\frac{1}{11}&2 < x\leq 5 \\ 0 &\text{otherwise} \end{cases}\]

  3. There are two ways to finish this. One is to use the original graph and from it we obtain \(P(X\leq 3) = \frac{8+1}{11} = \frac{9}{11}\) where the numerator is the area to the left of 3. The second option is to use the graph of \(f\), and then compute areas using it. Using the formula for \(f\), the area to the left of 3 is \((2)\frac{4}{11} + (1)\frac{1}{11}=\frac{9}{11}\).

  4. Calculate the expected value of \(X\). We have lots of options, either using \(g\) with the formula \(\frac{\sum x_i A_i}{\sum A_i}\), or use \(f\) with the formula \(\sum x_i A_i\). The normalized function \(f\) removes the need to divide by the total area.

Regardless of which option we choose, we must decide how many rectangles to use and then find the area and centroid of each. - We can use \(f\) with two rectangles. The left rectangle has area \(A_1 = 8/11\) and centroid at \(x_1=1\). The right rectangle has area \(A_2=3/11\) and centroid \(x_2 = 3.5\). We then compute \[E[X] = (1)\frac{8}{11} + (3.5)\frac{3}{11}\] - We can use \(f\) with 5 rectangles. Using equal sized rectangles makes this quick. The areas are A = c(4,4,1,1,1) centered at x = c(0.5,1.5,2.5,3.5,4.5). In R we can type sum(x*A) to rapidly compute the numerator and sum(A) to compute the denominator. \(\frac{\sum x_i A_i}{\sum A_i}\)

  1. To give a formula for \(F(x)\), we need to compute \(P(X\leq x)\). We can do this by calculating ratios of areas using either \(f\) or \(g\). Using \(g\), note that the area of a rectangle from 0 to \(x\) (for \(0\leq x\leq 2\)) is \(A=4x\). Dividing by the total area of 11 gives \(P(X\leq x) = \frac{4x}{11}\) for \(0\leq x\leq 2\). For \(2<x\leq 5\), we have a total area of \(A = 8+(x-2)*1 = x+6\), which means \(P(X\leq x) = \frac{x+6}{11}\) for \(2< x\leq 5\). This gives

    \[F(x) = \begin{cases}\frac{4x}{11}&0\leq x\leq 2\\\frac{x+6}{11}&2\leq x\leq 5\\0&x<0\\1&x>5\end{cases}\]

    Notice that plugging in 5 does gives us a probability of 1, as it should.

  2. The derivative of \(F\), where it is nonzero and defined, is \[\ds F'(x) = \begin{cases}\frac{4}{11}&0< x< 2\\\frac{1}{11}&2< x< 5 \end{cases}\] Notice again \(F'(x) = f(x)\).

Group Meeting

We’ll continue practicing using our new vocabulary.

  1. Let \(X\) represent the random variable obtained by recording the \(x\)-coordinate of a dart that is randomly thrown at a target that lies under the function \(\ds g(x) = \begin{cases}2x&0\leq x\leq 5\\0 &\text{otherwise} \end{cases}\).

    1. Draw the function.

  2. Find the area under \(g\).

  3. Give the normalized target function by finding the constant \(k\) so that the area under \(f(x) = kg(x)\) is 1.

  4. Compute \(P(X\leq 3)\).

  5. Calculate the expected value of \(X\). Use this Centroid Locations table.

  6. Give a formula for \(F(x)\), the cumulative distribution function.

  7. Compute \(F'(x)\).

Solutions

Compare your work with your group. Put agreed upon solutions on the chalkboard so you can compare with other teams.

  1. Let \(X\) represent the random variable obtained by recording the \(x\)-coordinate of a dart that is randomly thrown at a target that lies under the function \(\ds g(x) = \begin{cases}x^2&0\leq x\leq 5\\0 &\text{otherwise} \end{cases}\).

    1. Draw the function.

  2. Find the area under \(g\). Use this Centroid Locations table.

  3. Give the normalized target function by finding the constant \(k\) so that the area under \(f(x) = kg(x)\) is 1.

  4. Compute \(P(X\leq 3)\).

  5. Calculate the expected value of \(X\).

  6. Give a formula for \(F(x)\), the cumulative distribution function.

  7. Compute \(F'(x)\).

Activity - Calculating Expected Value using Rectangular Approximations

We computed expected values in the warm up today by referring to a list of geometric facts. What do we do if the region we’re interested in examining is not a simple geometric object? One option is to approximate the expect value using multiple rectangles. You’ll need to run the function below in R to complete the activity that follows.

draw_rect_approx() code 
draw_rect_approx <- function(f,a,b,num_rectangles, method = "mid"){
  n <- num_rectangles
  dx <- (b-a)/n
  x <- c(a,seq(a,b,dx/100),b,a)
  y <- c(0,f(seq(a,b,dx/100)),0,0)
  par(mar=c(2.5,2.5,0.25,0.25))
  plot(x,y,type = "l")

  if(method == "left"){
    xi <- seq(a+0*dx/2,b-dx/2,dx)
    lines(xi,f(xi),type = "h")
    lines(xi,f(xi),type = "s")
    lines(c(xi[n],xi[n]+dx),f(c(xi[n],xi[n])),type = "l")
    lines(c(xi[n],xi[n]+dx),f(c(xi[n],xi[n])),type = "h")
  }
  else if(method == "right"){
    xi <- seq(a+dx,b+dx/2,dx)
    lines(xi-dx,f(xi),type = "h")
    lines(xi-dx,f(xi),type = "s")
    lines(c(xi[n]-dx,xi[n]),f(c(xi[n],xi[n])),type = "l")
    lines(c(xi[n]-dx,xi[n]),f(c(xi[n],xi[n])),type = "h")
  } 
  else{#Use midpoint
    xi <- seq(a+dx/2,b,dx)
    lines(xi-dx/2,f(xi),type = "h")
    lines(xi-dx/2,f(xi),type = "s")
    lines(c(xi[n]-dx/2,xi[n]+dx/2),f(c(xi[n],xi[n])),type = "l")
    lines(c(xi[n]-dx/2,xi[n]+dx/2),f(c(xi[n],xi[n])),type = "h")
  }
}

For the function \(\ds g(x) = \begin{cases}2x&0\leq x\leq 5\\0 &\text{otherwise} \end{cases}\), the code below approximates the total area under \(g\) and expected value (of the corresponding random variable \(X\)) by treating the triangular target instead as 10 rectangular targets sewn together.

g <- function(x){2*x}
a <- 0
b <- 5
n <- 5
dx <- (b-a)/n
draw_rect_approx(g,a,b,n) 

xi <- seq(a+dx/2,b,dx)  #Center of each rectangle
Ai <- g(xi)*dx          #Area of each rectangle
data.frame(x_i = xi, A_i = Ai)

c(total_area =sum(Ai),  
  expected_value = sum(xi*Ai)/sum(Ai))

  1. Change the code above to use n=10 rectangular regions, then 50, then 100, then 1000. Discuss as a team why these approximations suggest that the expected value is \(\frac{10}{3} = 3.3\bar3\).

  2. Change the function above to \(\ds g(x) = \begin{cases}x^2&0\leq x\leq 5\\0 &\text{otherwise} \end{cases}\). Then use several values of n (start with small values and then increase them) to obtain approximations for the total area under \(g\) as well as the expected value of \(X\). Continue increasing the number of rectangular regions till you see the expected value is \(\frac{15}{4} = 3.75\). Note this is the same value we find if we use the table of centroids formula to find the expected value.

We’re ready to apply what we’re learning now to any function.

  1. Consider the target function \(\ds g(x) = \begin{cases}x^3&0\leq x\leq 4\\0 &\text{otherwise} \end{cases}\) and corresponding random variable \(X\) obtained by throwing darts at the target and recording the \(x\)-coordinate.
  • Find a constant \(k\) so that the area under \(f(x) = k g(x)\) is one (so find the normalized target function).
  • Compute \(P(X\leq 3)\) (you’ll need to compute two areas, and then divide them).
  • Calculate \(E[X]\)
  1. Consider the target function \(\ds g(x) = \begin{cases}5x e^{-x}&2\leq x\leq 8\\0 &\text{otherwise} \end{cases}\) and corresponding random variable \(X\) obtained by throwing darts at the target and recording the \(x\)-coordinate.
  • Find a constant \(k\) so that the area under \(f(x) = k g(x)\) is one (so find the normalized target function).
  • Compute \(P(X\leq 3)\) (you’ll need to compute two areas, and then divide them).
  • Calculate \(E[X]\)

Discussion

We’ll have a wrap up discussion.

Source: Class.35 on byuimath.com