readme: add exercise 4 and 5

README.md

@@ -34,15 +34,15 @@ libraries to build:
 * [pkg-config](https://www.freedesktop.org/wiki/Software/pkg-config/)
   (build-time only)
 
-Additionally Python (version 3) with `numpy` and `matplotlib` is required to
+Additionally, Python (version 3) with `numpy` and `matplotlib` is required to
 generate plots.
 
-For convenience a `shell.nix` file is provided to set up the build environment.
+For convenience, a `shell.nix` file is provided to set up the build environment.
 See this [guide](https://nixos.org/nix/manual/#chap-quick-start) if you have
 never used Nix before. Running `nix-shell` in the top-level will drop you into
 the development shell.
 
-Once ready, invoke `make` with the program you wishes to build. For example
+Once ready, invoke `make` with the program you wish to build. For example
 
     $ make ex-1/bin/main
 
@@ -115,6 +115,30 @@ Note: the histogram parameters affect the computation of the χ² and the
 relative parameter estimation.
 
 
+### Exercise 4
+
+`ex-4/bin/main` generates a sample of particles with random oriented momentum
+and creates an histogram with average vertical component, in modulus, versus
+horizontal component. It is possible to set the maximum momentum with the
+option `-p`. A χ² fit and a t-test compatibility are performed with respect
+to the expected distribution and results are printed.
+
+To plot a histogram of the generated sample do
+
+    $ ex-4/bin/main -o | ex-4/plot.py
+
+It is possible to set the number of particles and bins with the options `-n`
+and `-b`.
+
+
+### Exercise 5
+
+`ex-5/main` compute the integral of exp(1) between 0 and 1 with the methods
+plain MC, MISER and VEGAS. Being reiterative routines, it takes the number of
+iterations as its only argument.  
+It prints out the obatined value and its estimated error for each method.
+
+
 ### Exercise 6
 
 `ex-6/bin/main` simulates a Fraunhöfer diffraction experiment. The program
@@ -122,13 +146,13 @@ prints to `stdout` the bin counts of the intensity as a function of the
 diffraction angle. To plot a histogram simply pipe the output to the
 program `ex-6/plot.py`.
 
-The program convolves the original signal with a gaussian kernel (`-s` to
+The program convolves the original signal with a Gaussian kernel (`-s` to
 change the σ), optionally adds a Poisson noise (`-m` to change the mean μ) and
 performs either a naive deconvolution by a FFT (`-m fft` mode) or applying the
 Richard-Lucy deconvolution algorithm (`-m rl` mode), which is expected to
 perform optimally in this case.
 
-The `-c` and `-d` options controls whether the convolved or deconvolved
+The `-c` and `-d` options control whether the convolved or deconvolved
 histogram counts should be printed to `stdout`. For more options
 run the program with `-h` to see the usage screen.