While it's a bit strange to ask 'what percentage of results have 0.2 or fewer heads?' since we cannot get a partial number of heads, but it's easy to calculate that the only number of heads that is equal to or less than 0.2 is getting zero heads. COIN.cdf(2) asks 'what percentage of results have 2 or fewer heads?'.COIN.cdf(1) asks 'what percentage of results have 1 or fewer heads?'.COIN.cdf(0.2) asks 'what percentage of results have 0.2 or fewer heads?'.Using our two-coin flip example where COIN = binom(n=2, p=0.5), the CDF functions are asking the following: Graphically, this is the the total area of everything less than or equal to x (**the total area of the left of x*).

The probability of all outcomes less than or equal to a given value x,.The Cumulative Distribution Function or CDF is: We can represent this distribution as a table and a graph:

This is represented when COIN returns the value 2 ( exactly two heads). SciPy - Binomial Distribution Binomial Distribution is a discrete probability distribution and it expresses the probability of a given number of successes in a sequence of n independent experiments with a known probability of success on each trial.

There is a 25% probability of the outcome having two heads ( HH).

This is represented when COIN returns the value 1 ( exactly one head).

There is a 50% probability of the outcome having exactly one head ( TH or HT).

This is represented when COIN returns the value 0 ( zero heads).

There is a 25% probability of the outcome having zero heads ( TT).

The binomial distribution models these outcomes: There are four possible outcomes - HH, HT, TH, and TT. To create this distribution in Python: from scipy.

The functions are the same no matter what distribution you have - so let's discovery them via examples! Example Binomial DistributionР”Р„ simple binomial distribution that is easy to understand is a binomial distribution with n=2 and p=0.5 (two events, each with a 50% chance of success, like flipping a coin two times and finding out how many times we get heads). Once you have a variable with a distribution, there are many Python functions we can use to preform calculations with the distribution. Three extremely common distributions are normal, bernoulli and binomial distributions: DistributionР”В« = bernoulli (p = 0.2 ) # Bernoulli Distribution with p=0.2Р”В = binom (p = 0.5, n = 2 ) # Binomial Distribution with p=0.1, n=50 The variance is mu + mu^2/size in this parametrization.The scipy.stats library in Python provides us the ability to represent random distributions using Python! The library has dozens of distributions, including all commonly used distributions. The negative binomial distribution with size = n and prob = p has densityР”Р„n alternative parametrization (often used in ecology) is by the mean mu (see above), and size, the dispersion parameter, where prob = size/(size+mu). How can I use the mean/dispersion parametrization in scipy ? In R this is easy, as the negbin can be defined by both parametrizations: dnbinom(x, size, prob, mu, log = FALSE) However, there exists a different parametrization where the negative Binomial is defined by the mean mu and the dispersion parameter. This is the common definition, see also wikipedia: In scipy the negative binomial distribution is defined as: nbinom.pmf(k) = choose(k+n-1, n-1) * p**n * (1-p)**k

Scipy binomial