random.md

Random Sampling and Standard Distributions

1. check the functions in numpy.random

- https://docs.scipy.org/doc/numpy-1.16.0/reference/routines.random.html
- https://numpy.org/devdocs/reference/random/index.html

2. plot histograms

1. 1M samples generated from N(0,1)

   • np.random.normal()

2. 1M samples from a continuous uniform distribution U(0,1)

   • np.random.uniform()

3. 1M discrete random integer samples from U(0,1000)

   • np.randint()

4. 1M trials from binomial distribution with $n=100$ trials and the probability of success (or head) p = 0.3

   • $Pr[X=1] = 0.3 = p$
   • https://en.wikipedia.org/wiki/Binomial_distribution
   • np.random.binomial()

5. 1M samples from Bernoulli distribution with $P(X=1) = 0.3$

   • https://en.wikipedia.org/wiki/Bernoulli_distribution
   • the set of samples is composed of two numbers only, 0 and 1 (or A and B).
   • you can use np.random.binomial() or np.random.uniform() to generate samples from Bernoulli distribution.

6. (extra) How many times will a fair die land on the same number (e.g. 5) out of 100 trials.

   • use 'np.random.binomial(n=100, p=1/6., size=N)` to generate the samples.
   • base event set = { face is 5, face is not 5 }, so it is binary. (the same applies to other numbers each)

3. CDF (Cumulative distribution function), PDF (Prob. distribution function), PMF (Prob. Mass Function)

• https://en.wikipedia.org/wiki/Cumulative_distribution_function

1. plot cdf and pdf of Normal (or Gaussian) distribution with ($$mean=0$$, $$\sigma=1$$) and ($mean=0$, $\sigma=5$), respectively.
2. plot cdf and pdf of uniform distribution $U[0,1]$
3. plot cdf and pmf of Bernoulli distribution with $p=0.3$
4. plot cdf and pmf of $binomial(n=100, p=0.3)$