D$=1$# microns $-(2)/($s$)$; lengths are measured in microns

T$=5$#s

dt$=0\mathrm{.}02$#s $($same as $20$ms$)$

$[4]$: # three random walks

for $\_()$ in range$(3)$:

xs$,$ ys $=$ rw$\_(2)$d$($D$,$T$,$dt$)$

plt$.$plot$($xs$,$ys$)$

plt$.$gca$().$set$\_($a$)$spect$(1)$ #sets the aspect ratio

plt$.$xlabel$(\text{'}$x $($microns$)\text{'})$

plt$.$ylabel$(\text{'}$y $($microns$)\text{'})1.$ From trajectories to diffusion coefficients, part I: linear fit $(4$ points$)$

Let's call a set of $10$ independent trajectories with the parameters above one "experiment". From each experiment, we can calculate the MSD of the ten trajectories $\backslash (\backslash$left$\backslash$langle r$^\{2\}($t$)\backslash$right$\backslash$rangle $\backslash ),$ where $\backslash ($ r$^\{2\}=$x$^\{2\}+$y$^\{2\}\backslash )$ as before. Since the MSD$,$ for an extremely large number of trajectories, would approach $\backslash (4$ D t $\backslash ),$ a sensible estimate of D is $\backslash (\backslash$hat$\{$D$\}=\backslash )$ best$-$fit slope of $\backslash ($ r$^\{2\}\backslash )$ vs $\backslash ($ t $\backslash ),$ divided by $4.("$Best$-$fit" $=$ linear regression; in Python you can use scipy. stats.linregress. $($The 'hat' on the variable defined above signifies that it is an estimate of the true underlying $\backslash ($ D $\backslash )$ and in general will differ from $\backslash ($ D $\backslash ).)$ Write a program that uses this approach to estimate D$.$

i$.$ Suppose we only have resources to generate one experiment in the real world, with $10$ independent tracks measured in that experiment. We want to know how well the estimate from the linear fit above, applied to a single experiment, might match the true value of $\backslash ($ D $\backslash ).$ To do so$,$ simulate $400$ independent experiments $($each with $10$ independent trajectories$)$ and collect the estimate $\backslash (\backslash$hat$\{$D$\}\backslash )$ from each experiment. Generate and submit a histogram of the $\backslash (\backslash$hat$\{$D$\}\backslash )$ values $($you could use e$.$g$.$ matplotlib. pyplot $.$ hist; you can choose the number of bins to make sure most bins have a few hits atleast$).$ Also calculate and report the mean and standard deviation of the estimates across the $400$ experiments. How does the average $\backslash (\backslash$hat$\{$D$\}\backslash )$ from the $400$ experiments compare to the true value of $\backslash ($ D $\backslash )?$ If you only ran one experiment, your estimate would fall within one standard deviation of the estimated mean about $\backslash (68\backslash \%\backslash )$ of the time, so the standard deviation provides an estimation of how accurate your linear fit procedure would be$.$ ii$.$ A friend argues, "why don't you perform a linear fit of $\backslash ($ r$^\{2\}($t$)\backslash )$ against $\backslash ($ t $\backslash )$ for each of your ten trajectories, and then average the resulting ten estimates to get $\backslash (\backslash$hat$\{$D$\}\backslash )$ for the experiment? You're averaging over more fits within each experiment, so you would get better statistics." What is the flaw in this argument? Repeat part i$.,$ but now generate a $\backslash ($ D $\backslash )$ estimate from each experiment by perform ten fits for the individual trajectories and averaging over the $10$ resulting $\backslash (\backslash$hat$\{$D$\}\backslash )$ values. Is the spread of $\backslash ($ D $\backslash )$ estimates over the $400$ experiments better, worse, or the same as in i$.?$