We'll use a matrix to represent a bivariate or multivariate probability distribution and a vector to represent a univariate probability distribution

This {doc}`companion lecture <stats_examples>` describes some popular probability distributions and uses Python to sample from them.

This {doc}`companion lecture <stats_examples>` describes some popular probability distributions and describes how to use Python to sample from them.

In addition to what's in Anaconda, this lecture will need the following libraries:

$$

\textrm{Prob}\{X=i|Y=j\} =\frac{\textrm{Prob}\{X=i,Y=j\}}{\textrm{Prob}\{Y=j\}}=\frac{\textrm{Prob}\{Y=j|X=i\}\textrm{Prob}\{X=i\}}{\textrm{Prob}\{Y=j\}}

$$

$$ (eq:condprobbayes)

```{note}

This can be interpreted as a version of what a Bayesian calls **Bayes' Law**.

Formula {eq}`eq:condprobbayes` is also what a Bayesian calls **Bayes' Law**. A Bayesian statistician regards marginal probability distribution $\textrm{Prob}({X=i}), i = 1, \ldots, J$ as a **prior** distribution that describes his personal subjective beliefs about $X$.

He then interprets formula {eq}`eq:condprobbayes` as a procedure for constructing a **posterior** distribution that describes how he would revise his subjective beliefs after observing that $Y$ equals $j$.

```

$$ \textrm{Prob}(X=i)=\sum_j{f_{ij}}=u_i $$

$$ \textrm{Prob}(Y=j)=\sum_i{f_{ij}}=v_j $$

Below we draw some samples confirm that the "sampling" distribution agrees well with the "population" distribution.

**Sample results:**

**Sampling:**

Let's write some Python code that let's us draw some long samples and compute relative frequencies.

The code will let us check whether the "sampling" distribution agrees with the "population" distribution - confirming that

the population distribution correctly tells us the relative frequencies that we should expect in a large sample.

```{code-cell} ipython3

# specify parameters

print(x)

```

Here, we use exactly the inverse CDF technique to generate sample from the joint distribution $F$.

```{note}

To generate random draws from the joint distribution $F$, we use the inverse CDF technique described in {doc}`this companion lecture <stats_examples>`.

```

```{code-cell} ipython3

# marginal distribution

\end{array}\right]

$$

These population objects closely resemble sample counterparts computed above.

These population objects closely resemble the sample counterparts computed above.

Let's wrap some of the functions we have used in a Python class for a general discrete bivariate joint distribution.

Let's wrap some of the functions we have used in a Python class that will let us generate and sample from a

discrete bivariate joint distribution.

```{code-cell} ipython3

class discrete_bijoint:

plt.show()

```

Next we can simulate from a built-in `numpy` function and calculate a **sample** marginal distribution from the sample mean and variance.

Next we can use a built-in `numpy` function to draw random samples, then calculate a **sample** marginal distribution from the sample mean and variance.

```{code-cell} ipython3

μ= np.array([0, 5])

**Conditional distribution**

The population conditional distribution is

For a bivariate normal population distribution, the conditional distributions are also normal:

$$

\begin{aligned} \\

\end{aligned}

$$

```{note}

Please see this {doc}`quantecon lecture <multivariate_normal>` for more details.

```

Let's approximate the joint density by discretizing and mapping the approximating joint density into a matrix.

We can compute the discretized marginal density by just using matrix algebra and noting that

Thus, multiple joint distributions $[f_{ij}]$ can have the same marginals.

**Remark:**

- Couplings are important in optimal transport problems and in Markov processes.

- Couplings are important in optimal transport problems and in Markov processes. Please see this {doc}`lecture about optimal transport <opt_transport>`

## Copula Functions