We can take the matrix $A$= $\left( \begin{array}{cc} a & b \\ c & d \\\end{array} \right)$

and initial values $x_0, y_0$ placed into a vector $x_n$ = $\left( \begin{array}{c} x_0 \\ y_0 \end{array} \right)$

We then investigate repeated iterations of $Ax_n = x_{n+1}$

Analysis of the convergence of these matrices sees a connection to the direction of the matrix and a special number called an $eigenvalue (\lambda)$ and its corresponding $eigenvector$

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Another important, actually probably the most important, mathematical term introduced in this lab is randomized response. This is a research method used in structured survey interview. This method allows respondents to respond to sensitive issues while maintaining confidentiality. Chance decides, unknown to the interviewer, whether the question is to be answered truthfully, or “yes”, regardless of the truth.

A question I’m particularly curious about involves finding a formula to estimate the proportion of the true “yes” answers.

Using math formulas in such a practical way and trying to apply logic and structure to the chaos that is human response and just human interaction is very intriguing to me.

]]>$y=f(x)$ or $z=f(x,y)$,

parametric curves give the values for x and y as a function of t. So:

$x=x(t)$ and $y=y(t)$

This way of representing curves can be very useful with circles and polar coordinates. The questions that they ask deal mostly with parameters of $sin(at)$ and $cos(bt)$. The first few questions start exploring functions and seeing what they do with different values, how they intersect, etc. And then forming conjectures based on the data.

]]>Human psychology, as I have said in my earlier post on randomized response, is very interesting to me. I appreciate the difficulty in conducting an accurate survey in order to study human behavior. In this lab we are asked to explore different aspects in randomized response such as the margin of error and probabilities. I think that it would be incredibly interesting to conduct our own survey using the randomized response method and use that to study the different aspects of the method that the lab talks about. I would also just like to better understand the process of conducting a randomized response survey in order to be able to utilize it in the future.

]]>As we can all remember from Trigonometry, the sine and cosine functions oscillate in a orderly fashion when we look at it graphically. By orderly fashion, I mean to say that we know the trend of the oscillation after observing it for a short period of time. We can also combine them, take the derivative, and integrate such functions to make a whole new function.

For this lab, we observe such periodic functions by calculating the area under the functions. In other words, we evaluate the definite integrals.

]]>There is a little twist, though. Iteration around a fixed may cause nearby points to converge towards that fixed point. That point is then called an *attractor. *Nearby points could also diverge away from the fixed point. If so, the point is then called a *repeller.* Sometimes there is no pattern at all and can look quite chaotic……..

Chaos will be looked at and that is the major draw for me to consider this chapter. Let’s have some random fun!!!

]]>Before, we were iterating the function $f(x) = ax+b$. This sort of iteration is called an **affine** map. In this lab we will be doing **linear** maps or maps where the constant term $b = 0$. Further, because we are in the plane, our function will have vector valued inputs and will have vector-valued output. So our function will look something like:

$$ f(x, y) = (a_{11}x + a_{12}y, a_{21}x + a_{22}y) $$

Or similarly, we can write our equation:

$$

\left({}

\begin{array}{c}

x_{n+1} \\{}

y_{n+1}

\end{array}

\right){}

=

f(x_n, y_n)

=

A

\left({}

\begin{array}{c}

x_n \\{}

y_n

\end{array}

\right){}

$$

where $ A = \left({}

\begin{array}{c}

a_{11} & a_{12} \\{}

a_{21} & a_{22}

\end{array}

\right){} $.

Some questions this lab seeks to answer are similar to our first lab: it will ask us to try some different variations of our matrix $A$ and/ or our initial values and see if we can notice a pattern.

]]>One of the questions involves finding a function for a particular survey. By using a proportion concerning the number of yes answers versus the number of total answers, it is asked to find an equation that can help in estimating what the true number of yeses should be to the real question. Further questions build on determining how to uncover accurate results using the probabilities in possible answers from the real question and the decoy question.

I think this is a very useful lab for statistical analysis because you want people to be honest when gathering data but you simply cannot rely on honesty. Finding an accurate way to collect data while helping others to feel comfortable in being honest helps in gathering accurate data.

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