Here’s the solution:

```
# Exercise 6: Fit linearised Radioactive Decay
# Transform data and parameters to linear form: Y = A + B*X
X = t # transform t data for fitting (trivial)
Y = log(N) # transform N data for fitting
dY = dN/N # transform uncertainties for fitting
def linearfit(x,m,c):
return m*x+c
popt,pcov = curve_fit(linearfit, X, Y, sigma=dY)
sigma = sqrt(diag(pcov))
slope = popt[0]
tau = -1/slope
dtau = tau**2 * sigma[0]
intercept= popt[1]
Nfit = exp(intercept)
dNfit = Nfit * sigma[1]
print Nfit,dNfit
print tau,dtau
xfit = linspace(0, 500, 100)
yfit = slope*xfit + intercept
# Plot Data
errorbar(X, Y, fmt='o',label='Data',yerr=dY)
plot(xfit,yfit, label='Weighted Linear Fit')
xlabel("time [days]")
ylabel("ln($N$)")
# Add Legend
legend(loc='best') # best location to avoid hiding data
# Save figure as png file with 300dpi
savefig('DecayLinearisedFit.png', dpi=300, format='png', bbox_inches='tight')
```

## So what’s going on ?

After creating variables for convenience, we *define a new function* called `linearfit`

. Notice the syntax:

```
def function_name(<arguments>):
<operations>
return <output>
```

In our case, we don’t need this function to do anything else than evaluate a simple expression, so we can just set it to *return* a number. Next we’re calling the `curve_fit`

method we loaded from `scipy.optimize`

(see here). This is an algorithm that uses non-linear least squares to fit our function `linearfit`

to the data arrays `X`

and `Y`

. It assumes that `Y`

is a function of `X`

with some parameters to adjust and (possibly) an uncertainty `sigma`

(offset). It returns an array, `popt`

, which contains the optimised parameters of `linearfit`

(`m`

and `c`

) such as to minimize

which is exactly what a fit is. The algorithm also returns a 2d-array `pcov`

, which you can think of as a matrix. The diagonal contains the variance of the `popt`

, such that we can extract the one standard deviation errors by taking the square root of those elements. Note that the way we define `sigma`

makes it a 1d-array.

The rest of the exercise should by now be familiar and quite straightforward.

*Note: if you’re wondering how curve_fit knows what the parameters and the x-variable are when we only pass linearfit as argument, you’re starting to ask yourself the right questions. Kudos. In fact, curve_fit assumes that the function you pass as argument is of the form function(x,*parameters), which is exactly why we ordered the arguments of linearfit in its definition the way we did.*