Product Distributions

Why

How do we get a joint probability distribution function from marginals?¹

Definition

The product distribution of a sequence $p_1, \dots , p_n: A_i \to [0, 1]$ of distributions is the function $p: \prod_{i} A_i \to [0, 1]$ defined by

\[ p(x) = \prod_{i = 1}^{n} p_i(x_i). \]

Let $p_i: A_i \to [0, 1]$ be probability distributions. Then the product distribution of $p_1, \dots , p_n$ is a probability distribution with marginals $p_1, \dots , p_n$.²

Example: fair coin repeated flips

Suppose we want to model a coin flipped $n$ times. Supposing the coin is fair (see \sheetref{probability_distribtuions}{Probability Distributions}) we might use our probability distribution $p: \set{0, 1} \to [0, 1]$ which assigned $p(0) = p(1) = \nicefrac{1}{2}$. Then the probability of obtaining a sequence of flips $x \in \set{0, 1}$ is

\[ \textstyle p(x) = \prod_{x_i = 1} p_i(1) \prod_{x_i = 0} p_i(0). \]

Notice that if we know $p_i(1) = \rho _i$, then we know $p(0) = (1-\rho _i)$ and so we can write the above as

\[ \textstyle p(x) = \prod_{x_i = 1} \rho _i \prod_{x_i = 0} (1-\rho _i). \]

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