\displaystyle \textbf{1.}\ \text{Let }S\text{ be the sample space associated with a given random experiment. Then, a} \\ \text{real valued function }X\text{ which assigns to each event }\omega\in S\text{ a unique real number }\\ X(\omega)\text{ is called a random variable.}
\displaystyle \text{In other words, a random variable is a real valued function having domain as the sample} \\ \text{space associated with a random experiment.}

\displaystyle \textbf{2.}\ \text{If a random variable }X\text{ takes values }x_{1},x_{2},\ldots,x_{n} \\ \text{ with respective probabilities }p_{1},p_{2},\ldots,p_{n},\text{ then}
\displaystyle \begin{array}{c|ccccc} X & x_{1} & x_{2} & x_{3} & \cdots & x_{n}\\ \hline \mathrm{P}(X) & p_{1} & p_{2} & p_{3} & \cdots & p_{n} \end{array}
\displaystyle \text{is known as the probability distribution of }X.

\displaystyle \textbf{3.}\ \text{The probability distribution of a random variable }X\text{ is defined only when} \\ \text{we have the various values }x_{1},x_{2},\ldots,x_{n}\text{ together with respective probabilities } p_{1},p_{2},\ldots,p_{n} \\ \text{ satisfying } \sum_{i=1}^{n}p_{i}=1.

\displaystyle \textbf{4.}\ \text{If }X\text{ is a random variable with the probability distribution }
\displaystyle \begin{array}{c|cccc} X & x_{1} & x_{2} & \cdots & x_{n}\\ \hline \mathrm{P}(X) & p_{1} & p_{2} & \cdots & p_{n} \end{array}
\displaystyle \text{then,}
\displaystyle \mathrm{P}(X\leq x_{i})=\mathrm{P}(X=x_{1})+\mathrm{P}(X=x_{2})+\cdots+\mathrm{P}(X=x_{i})=p_{1}+p_{2}+\cdots+p_{i}.
\displaystyle \mathrm{P}(X<x_{i})=\mathrm{P}(X=x_{1})+\mathrm{P}(X=x_{2})+\cdots+\mathrm{P}(X=x_{i-1})=p_{1}+p_{2}+\cdots+p_{i-1}.
\displaystyle \mathrm{P}(X\geq x_{i})=\mathrm{P}(X=x_{i})+\mathrm{P}(X=x_{i+1})+\cdots+\mathrm{P}(X=x_{n})=p_{i}+p_{i+1}+\cdots+p_{n}.
\displaystyle \mathrm{P}(X>x_{i})=\mathrm{P}(X=x_{i+1})+\mathrm{P}(X=x_{i+2})+\cdots+\mathrm{P}(X=x_{n})=p_{i+1}+p_{i+2}+\cdots+p_{n}.
\displaystyle \text{Also, } \mathrm{P}(X\geq x_{i})=1-\mathrm{P}(X<x_{i}),\ \mathrm{P}(X>x_{i})=1-\mathrm{P}(X\leq x_{i}).
\displaystyle \mathrm{P}(X\leq x_{i})=1-\mathrm{P}(X>x_{i}),\ \mathrm{P}(X<x_{i})=1-\mathrm{P}(X\geq x_{i}).
\displaystyle \mathrm{P}(x_{i}\leq X\leq x_{j})=\mathrm{P}(X=x_{i})+\mathrm{P}(X=x_{i+1})+\cdots+\mathrm{P}(X=x_{j}).
\displaystyle \mathrm{P}(x_{i}<X<x_{j})=\mathrm{P}(X=x_{i+1})+\mathrm{P}(X=x_{i+2})+\cdots+\mathrm{P}(X=x_{j-1}).

\displaystyle \textbf{5.}\ \text{If }X\text{ is a discrete random variable which assumes values } x_{1},x_{2},x_{3},\ldots,x_{n} \\\text{with respective probabilities }p_{1},p_{2},\ldots,p_{n},\text{ then the mean } \\ \overline{X}\text{ of }X\text{ is defined as}
\displaystyle \overline{X}=p_{1}x_{1}+p_{2}x_{2}+\cdots+p_{n}x_{n}=\sum_{i=1}^{n}p_{i}x_{i}.
\displaystyle \text{The mean of a random variable }X\text{ is also known as its mathematical expectation or expected} \\ \text{value and is denoted by }E(X).

\displaystyle \textbf{6.}\ \text{If }X\text{ is a discrete random variable which assumes values }x_{1},x_{2},x_{3},\ldots,x_{n} \\ \text{ with respective probabilities }p_{1},p_{2},\ldots,p_{n},\text{ then the variance of } \\ X\text{ is defined as}
\displaystyle \mathrm{Var}(X)=p_{1}(x_{1}-\overline{X})^{2}+p_{2}(x_{2}-\overline{X})^{2}+\cdots+p_{n}(x_{n}-\overline{X})^{2}=\sum_{i=1}^{n}p_{i}(x_{i}-\overline{X})^{2},
\displaystyle \text{where }\overline{X}=\sum_{i=1}^{n}p_{i}x_{i}\text{ is the mean of }X.
\displaystyle \text{Also, }\mathrm{Var}(X)=\sum_{i=1}^{n}p_{i}x_{i}^{2}-\left(\sum_{i=1}^{n}p_{i}x_{i}\right)^{2}.

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