2 Basic Properties of Random Vectors

Let \(\underline {X}'=\begin {pmatrix} x_1, & x_2, & \cdots , & x_p\\ \end {pmatrix} \) be a random vector. Then

1.

The cumulative distribution function (CDF) is the function \(F\) defined by \(F(\underline {X}')\) \begin {equation} \tag {1} F(\underline {X}^0)=Pr\big (\underline {X}'\leq \underline {X}^0\big ) = Pr\big (x_1\leq x_1^0, x_2\leq x_2^0, \ldots , x_p\leq x_p^0\big ) \end {equation}

\[Pr(\underline {X}\leq x_2^0) = Pr(x_1\leq x_1^0, x_2\leq x^0_2)\]

There are two cases continuous and discrete distributions.

(a)

\(\displaystyle {F(\underline {X} = \int ^{\infty }_{-\infty }f(\underline {U})d\underline {U}}\qquad (2)\) where \(d\underline {U}=du_1\cdot du_2\cdot \cdots du_p\) \[\int ^{\underline {X}}_{-\infty } = \int ^{x_1}_{-\infty }\int ^{x_2}_{-\infty }\cdots \int ^{x_p}_{-\infty }\]

(b)

\(\displaystyle {\int ^{\infty }_{-\infty }f\big (\underline {U}\big )d\underline {U}=1}\)

A random vector \(\underline {X}\) is discrete if there exists a probability mass function (Pmf) \(f\big (\underline {U}\big )\) such that \[f\big (\underline {X}_j\big ) = \begin {cases} Pr\big (\underline {X}=\underline {X}_j & j=1,2,\ldots \\\\ 0, & \text {otherwise}\\ \end {cases} \]

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