91Ó°ÊÓ

Consider a discrete random variable \(X\) with the following pmf: $$ p_X(x)= \begin{cases}\frac{1}{x(x+1)}, & x=1,2, \ldots \\ 0, & \text { otherwise }\end{cases} $$ cant copy graph Figure 4.P.1. An alternative method of computing \(E[X]\) Show that the function defined satisfies the properties of a pmf. Show that the formula (4.1) of expectation does not converge in this case and hence \(E[X]\) is undefined. [Hint: Rewrite \(1 / x(x+1)\) as \(1 / x-1 /(x+1) \cdot]\)

The time to failure distribution of Tandem software was found to be captured well by a two-phase hyperexponential distribution with the following pdf: $$ f(t)=\alpha_1 \lambda_1 e^{-\lambda_1 t}+\alpha_2 \lambda_2 e^{-\lambda_2 t}, $$ with \(\alpha_1=0.87, \alpha_2=0.13, \lambda_1=0.10, \lambda_2=2.78\) [LEE 1993]. Find the mean and variance of the time to failure.

Consider random variables \(X\) and \(Y\) with the joint pdf (bivariate Gaussian): $$ \begin{aligned} f(x, y) &=\frac{1}{2 \pi \sigma_X \sigma_Y \sqrt{1-\rho^2}} \\ & \cdot \exp \left\\{-\frac{1}{2\left(1-\rho^2\right)}\left[\left(\frac{x-\mu_X}{\sigma_X}\right)^2-\frac{2 \rho\left(x-\mu_X\right)\left(y-\mu_Y\right)}{\sigma_X \sigma_Y}+\left(\frac{y-\mu_Y}{\sigma_Y}\right)^2\right]\right\\} \end{aligned} $$ where \(\rho \neq \pm 1\). Show that \(\operatorname{Cov}(X, Y)=\rho \sigma_X \sigma_Y\). Hence show that if \(X, Y\) are jointly Gaussian and uncorrelated (i.e., \(\rho=0\) ), then they are also independent. Note that this is not true in general.