proposition (the linearity of expectation)let and be two random variables and let be number then ( + ( ( and (cx ce( example let be random variable that assigns the outcome of the roll of two fair dice to the sum of the number of dots showing then ( justificationto justify this claimlet and be random variables corresponding to the number of dots on each die thusx ( they are two instances of the same functionand ( ( ( ( each outcome of the roll of fair die occurs with probability / thuse(xi for thereforee( two random variables and are independent if pr( | ypr( )for all real numbers and proposition if two random variables and are independentthen (xy ( ) ( example let be random variable that assigns the outcome of roll of two fair dice to the product of the number of dots showing then ( / justificationlet and be random variables denoting the number of dots on each die the variables and are clearly independenthence ( ( ( ) ( ( / ) / the following bound and corollaries that follow from it are known as chernoff bounds proposition let be the sum of finite number of independent / random variables and let be the expected value of thenfor ed pr( ( ) ( )( +