Joint Distribution¶
The joint distribution describes the probability of two or more random variables occurring simultaneously. Joint distribution of \(\mathbf{X}\) and \(\mathbf{Y}\) in discrete case, defined as :
then,
- \(P(\mathbf{X}=x, \mathbf{Y}=y) \ge 0\)
- \(\sum_{x,y} P(\mathbf{X}=x, \mathbf{Y}=y)=1\)
- The CDF of joint distribution are \(P[(x,y)\in A] = \int_A f(x, y) \; dxdy\)
Each individual distribution within the joint distribution is called the marginal distribution:
- \(P(\mathbf{X}=x) =\sum_y P(\mathbf{X}=x)\)
- \(P(\mathbf{Y}=Y) =\sum_x P(\mathbf{Y}=y)\)
Both called marginal distribution of \(\mathbf{X}\) and \(\mathbf{Y}\), and the marginal density is defined as:
- \(f_\mathbf{X}(x) = \int_{-\infty}^{\infty} f(x,y) \; dy\)
- \(f_\mathbf{Y}(y) = \int_{-\infty}^{\infty} f(x,y) \; dx\)
where \(f(x,y)\) is the joint PDF. The conditinal distributions of \(\mathbf{X}\) and \(\mathbf{Y}\),
and
Independence of Joint Probability¶
Independence of joint probability same as single probability. Multiple random variables \(X_1, X_2, X_3, \cdots X_n\) are independent, if for all \((x_1, x_2, x_3, \cdots x_n)\in \mathbb{R}^n\).
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If \(X_1, X_2, X_3, \cdots X_n\) are discrete, then
\[ P(X_1=x_1, X_2=x_2, X_3=x_3, \cdots X_n=x_n) = P(X_1=x_1)P(X_2=x_2)P(X_3=x_3)\cdots P(X_n=x_n) \] -
If \(X_1, X_2, X_3, \cdots X_n\) are continous, then
\[ f(X_1=x_1, X_2=x_2, X_3=x_3, \cdots X_n=x_n) = f(X_1=x_1)f(X_2=x_2)f(X_3=x_3)\cdots f(X_n=x_n) \] -
If \(X_1, X_2, X_3, \cdots X_n\) are independent and identically distribute (IID), then they have same CDF, the same means, and the same variances