Faisal Shahzad, 30, a United States naturalized citizen from Pakistan, pleaded guilty to a failed attempt to explode a car bomb in Times Square in New York City in May 2010. Shahzad was sentenced to life in prison. (Songquan Deng/Shutterstock.) $z\u2010\mathrm{value}\text{}1.645$ Faisal Shahzad, 30, a United States naturalized citizen from Pakistan, pleaded guilty to a failed attempt to explode a car bomb in Times Square in New York City in May 2010. Shahzad was sentenced to life in prison. (Songquan Deng/Shutterstock.)

$$P\left(\mathit{ppf}\right)=\left(0.8\right)\left(0.8\right)\left(0.2\right)=0.128$$

$$\begin{array}{c}\left(1\right)\text{}f\left({x}_{i}\right)\ge 0\hfill \\ \left(2\right)\text{}{\displaystyle \sum _{i=1}^{n}f\left({x}_{i}\right)}=1\hfill \\ \left(3\right)\text{}f\left({x}_{i}\right)=P\left(X={x}_{i}\right)\hfill \end{array}$$

$$f\left(x\right)=\left\{0.038,,,x,=,1,0.102,,,x,=,2,0.172,,,x,=,3,0.204,,,x,=,4,0.174,,,x,=,5,0.124,,,x,=,6,0.080,,,x,=,7,0.036,,,x,=,8,0.028,,,x,=,9,0.022,,,x,=,10,0.020,,,x,=,15\right.$$

Formula 1: $$ \begin{equation} \label{Formula1.Label} P (\Omega) = \int_{-\infty}^{+\infty} f (x) \: dx = 1 \end{equation} $$

Formula 2: $$ \begin{equation} \label{Formula2.Label} E (X) := \int_{-\infty}^{+\infty} x \: f (x) \; dx \end{equation} $$

A third one, with references to 1 and 2: $$ \begin{equation} \begin{array}{ll} E (a X + b) & = \int_{-\infty}^{+\infty} (a x - b) \: f(x) \: dx \\ \\ & = a \int_{-\infty}^{+\infty} x \: f(x) \: dx + b \int_{-\infty}^{+\infty} f(x) \: dx \\ \\ & \underset{\eqref{Formula1.Label}, \eqref{Formula2.Label}}{=} a E(X) + b \end{array} \end{equation} $$

And another one: $$ \begin{equation} V(X) = \frac{a^2 + b^2 + c^2 - ab - ac - bc}{18} \end{equation} $$