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Probability and Statistics


Mathematical Expectation


  • Let XX be a discrete random variable with PMFPMF pXp_X and support SXS_X, and if xSXxpX(x)<\sum_{x\in S_X}|x|p_X(x)<\infty, then we say that expectation of XX exists, defined as xSXxpX(x)\sum_{x\in S_X}xp_X(x) (i.e Expectation exists if xSxxpX(x)\sum_{x\in S_x}xp_X(x) converges absolutely)
  • Let XX be a continuous random variable with PDFPDF fX(x)f_X(x) and if xfX(x)dx<\int_{-\infty}^{\infty}|x|f_X(x)dx<\infty, then we say expectation exists, and defined as xfX(x)dx\int_{-\infty}^{\infty}xf_X(x)dx (i.e Expectation exists if xfX(x)dx\int_{-\infty}^{\infty}xf_X(x)dx (as improper integral)converges absolutely)