1.8 Some Measures on Random Variable
From the ECE600 Pre-requisites notes of Sangchun Han, ECE PhD student.
1.8.1 Mean
$ E\left[\mathbf{X}\right]=\sum_{k}k\cdot p_{\mathbf{X}}\left(k\right)=\int_{-\infty}^{\infty}x\cdot f_{\mathbf{X}}\left(x\right)dx $.
1.8.2 Variance
$ Var\left[\mathbf{X}\right]=E\left[\left(\mathbf{X}-E\left[\mathbf{X}\right]\right)^{2}\right]=E\left[\mathbf{X}^{2}\right]-\left(E\left[\mathbf{X}\right]\right)^{2} $.
$ E\left[\mathbf{X}^{2}\right]=\left(E\left[\mathbf{X}\right]\right)^{2}+Var\left[\mathbf{X}\right] $.
$ \sigma_{\mathbf{X}}=\sqrt{Var\left[\mathbf{X}\right]} $.
1.8.3 Covariance
$ Cov\left(\mathbf{X},\mathbf{Y}\right) $
1.8.4 Correlation coefficient
$ r=\rho_{\mathbf{XY}}=\frac{Cov\left(\mathbf{X},\mathbf{Y}\right)}{\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}}=\frac{E\left[\mathbf{XY}\right]-E\left[\mathbf{X}\right]E\left[\mathbf{Y}\right]}{\sigma_{\mathbf{X}}\sigma_{\mathbf{Y}}} $.
• This range from -1.0 to 1.0 because of Cauchy–Schwarz inequality.
1.8.5 Characteristic function
$ \Phi_{\mathbf{X}}(\omega)=E\left[e^{i\omega\mathbf{X}}\right]=\int_{-\infty}^{\infty}f_{X}(x)e^{i\omega x}dx=\sum_{k=0}^{\infty}p_{X}\left(k\right)e^{i\omega k} $
$ E\left[\mathbf{X}\right]=\int_{-\infty}^{\infty}x\cdot f_{\mathbf{X}}\left(x\right)dx=\frac{d}{d\left(i\omega\right)}\Phi_{\mathbf{X}}\left(\omega\right)\left|_{i\omega=0}\right. $
$ E\left[\mathbf{X}^{2}\right]=\int_{-\infty}^{\infty}x^{2}\cdot f_{\mathbf{X}}\left(x\right)dx=\frac{d}{d\left(i\omega\right)^{2}}\Phi_{\mathbf{X}}\left(\omega\right)\left|_{i\omega=0}\right. $
