Distributions: Primitives and Combinations

In this section, we will build up some primitive distributions, as well as one means of combining them. See section 8.5 of the readings for information and examples of the types of distributions below.

For each of the boxes below, assume that DDist, make_joint_distribution, and total_probability are available.

Square Distribution

Define a function square_dist(lo, hi, loLimit=None, hiLimit=None) that returns an instance of DDist representing a distribution that is uniform across integer elements from lo to hi-1, inclusive. Any probability mass that would be associated with elements below loLimit or above hiLimit is assigned to loLimit or hiLimit, respectively.

def square_dist(lo, hi, loLimit = None, hiLimit = None): pass # your code here

Triangle Distribution

Define a function triangle_dist(peak, halfWidth, loLimit, hiLimit) that returns an instance of DDist representing a distribution over integers whose probability mass has its peak at peak and falls off linearly from there, reaching probability 0 at peak+halfWidth and peak-halfWidth. As with square_dist, any probability mass that would be associated with elements below loLimit or above hiLimit should be assigned to loLimit or hiLimit, respectively.

def triangle_dist(peak, halfWidth, loLimit = None, hiLimit = None): pass # your code here

Mixture

We can make a new distribution by defining it to be a mixture of two existing distributions. By specifying two distributions d1 and d2 and a mixture probability p, the mixture distribution assigns to each element x a probability equal to p times the probability of x in distribution d1 plus (1-p) times the probability of x in distribution d2.

Implement a function mixture(d1,d2,p), which returns an instance of DDist representing the mixure of d1 and d2 with mixing probability p.

def mixture(d1, d2, p): pass # your code here