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Probability

# Probability - University of Central Florida School of...

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University of Central Florida School of Electrical Engineering and Computer Science COP 4600 - Operating Systems Spring 2011 - dcm Probability and Statistics Concepts Random Variable: a rule that assigns a numerical value to each possible outcome of an ex- periment. All possible outcomes of the experiment constitute a sample space . A random variable X on a sample space S is a function X : S 7→ R which assigns a real number X ( s ) to every sample point s S . This real number is called the probability of that outcome. A discrete random variable maps events to values of a countable set e.g., the set of integers); each value in the range has a probability greater than or equal to zero. Example 1. the experiment is a coin toss; the outcome is either 0 (head) or 1 (tail). If the coin is fair then p 0 = p 1 = 0 . 5; this means that in a large number of coin tosses we are likely to observe heads in about half of the cases and tails in the other half of the cases. Another example: when you throw throw a dice the outcome could be 1 , 2 , 3 , 4 , 5, or 6; for a fair dice p 1 = p 2 = p 3 = p 4 = p 5 = p 6 = 1 / 6. A continuous random variable maps events to values of an uncountable set (e.g., the real numbers). Example 2. the experiment is to measure the speed of cars passing through an intersection: the speed could be any value between 15 and 80 miles/hour. the probability of observing cars with a speed of 19 . 1 miles/hour could be zero but the probability of observing cars with a speed from 15 to 19 . 1 miles/hour could be P 19 . 1 = 0 . 3 which means that 30% of the cars we observed have a speed in the range we considered. A discrete random variable X has an associated probability density function , (also called probability mass function) p X ( x ) defined as: p X ( x ) = Prob( X = x ) and a probability distribution function also called cumulative distribution function , P X ( x ) defined as: P X ( t ) = Prob( X t ) = X x t p X ( x ) Example 3. You have a binary random variable X (the outcome is either 0 or 1 ) and: p 0 = Prob( X = 0) = q and p 1 = Prob( X = 1) = p, with p + q = 1 . Bernouli trials: call the outcome of 1 a “success” and ask the question what is the probability Y n that in n Bernoulli trials we have k successes: p k = Prob( Y n = k ) = n k p k (1 - p ) n - k = n ! k !( n - k )! p k (1 - p ) n - k

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The binomial cumulative distribution function is: B ( t : n, p ) = t X k =0 n k p k (1 - p ) n - k Example 4. You have again Bernoulli trials and ask the question how many trails you need before the first “success”. If the first success occurs at the i -th trial then p
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