Linear algebra viewpoint of function of discrete random variable

Suppose X is a discrete random variable can take on n values x_{1} < x_{2} <\ldots <x_{n} with probability p_{x_{1}},p_{x_{2}},\ldots ,p_{x_{n}}; and Y is a random variable defined by Y = g(X).

Define a column vector \lambda_{X} = [p_{x_{1}} \quad p_{x_{2}} \quad \ldots \quad p_{x_{n}}]^{T}, can be called the probability mass distribution vector (?) of X

Define a n \text{x} n matrix P = \left[ \begin{array}{c} a_{1} \\ a_{2} \\ \ldots \\ a_{n} \end{array}\right] with a_{1},a_{2},\ldots ,a_{n} are 1\text{x}n row vectors.
P can be called the mapping probability mass matrix (?). The values of P is based on the sample space values x_{1},x_{2},\ldots ,x_{n} and the function g.
Then \lambda_{Y} -the probability mass distribution vector of Y– can be obtained as followed:
\lambda_{Y}=P\cdot \lambda_{X} = \left[ \begin{array}{c} a_{1} \\ a_{2} \\ \ldots \\ a_{n} \end{array}\right] \cdot \left[ \begin{array} {cccc} p_{x_{1}} & p_{x_{2}} & \ldots & p_{x_{n}} \end{array}\right] ^{T}


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