Easy

Have a go at it. The editor and the docs panel are open, and your code is saved as you type. Running it needs a free account — you’ll come back to exactly what you wrote.

Marginal and Conditional Probability

~10 mincode completion

A joint distribution $P (X, Y)$ is a table: entry $(i, j)$ is the probability of $X = i$ and $Y = j$ . The whole table sums to 1.

Two things you can extract from it:

Marginal — collapse one variable by summing it away.

That is a row sum, which is joint.sum(axis=1). The name comes from writing the totals in the margin of the table.

Conditional — restrict to one row and renormalise so it sums to 1 again.

P (Y = j ∣ X = i) = \frac{P ( X = i , Y = j )}{P ( X = i )}

joint = [[0.1, 0.2],
         [0.3, 0.4]]

row sums (marginal over Y): [0.3, 0.7]
P(Y | X=0) = [0.1, 0.2] / 0.3 = [0.3333, 0.6667]

Conditioning is where axis and keepdims stop being trivia: divide a (n, m) table by a (n,) vector and numpy broadcasts down the columns instead of across the rows, and you get numbers rather than an error.

Your task:

Implement conditional_given_row(joint, row) returning $P (Y ∣ X = row)$ as a 1-D array that sums to 1.

Example Tests

Conditioning on the first row renormalises [0.1, 0.2] to sum to 1

Input: {"row":0,"joint":[[0.1,0.2],[0.3,0.4]]}

Expected: [0.33333,0.66667]

The second row is already twice as likely at its right-hand entry

Input: {"row":1,"joint":[[0.1,0.2],[0.3,0.4]]}

Expected: [0.42857,0.57143]

A row that is already uniform stays uniform

Input: {"row":0,"joint":[[0.25,0.25],[0.25,0.25]]}

Expected: [0.5,0.5]

Python

import numpy as np


def conditional_given_row(joint, row):
    """
    Conditional distribution of Y given a fixed value of X.

    Args:
        joint: 2-D array of joint probabilities, shape (n, m), summing to 1
        row:   which value of X to condition on

    Returns:
        array of shape (m,) summing to 1
    """
    # YOUR CODE HERE
    pass

Loading docs…