discrete random variables

discrete random variables - Example In n trials let X be...

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Example. In n trials, let X be the number of successes. A discrete random variable is one that can only take countably many values. For a discrete random variable, we define the probability mass function or the density by p ( x ) = P ( X = x ). Here P ( X = x ) is an abbreviation for P ( { ω S : X ( ω ) = x } ). This type of abbreviation is standard. Note i p ( x i ) = 1 since X must equal something. Let X be the number showing if we roll a die. The expected number to show up on a roll of a die should be 1 · P ( X = 1) + 2 · P ( X = 2) + · · · + 6 · P ( X = 6) = 3 . 5. More generally, we define E X = { x : p ( x ) > 0 } xp ( x ) to be the expected value or expectation or mean of X . Example. If we toss a coin and X is 1 if we have heads and 0 if we have tails, what is the expectation of X ? Answer. p X ( x ) = 1 2 x = 1 1 2 x = 0 0 all other values of x . Hence E X = (1)( 1 2 ) + (0)( 1 2 ) = 1 2 . Example. Suppose X = 0 with probability 1 2 , 1 with probability 1 4 , 2 with probability 1 8 , and more generally n with probability 1 / 2 n . This is an example where X
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