L17_RandomVariables_print

L17_RandomVariables_print - 1 COMP170 Discrete Mathematical...

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Unformatted text preview: 1 COMP170 Discrete Mathematical Tools for Computer Science Discrete Math for Computer Science K. Bogart, C. Stein and R.L. Drysdale Section 5.4, pp. 249-262 Random Variables Version 2.0: Last updated, May 13, 2007 Slides c 2005 by M. J. Golin and G. Trippen 2 Random Variables • What Are Random Variables? • Binomial Probabilities • Expected Values • Expected Values of Sums and Numerical Multiples • Indicator Random Variables • The Number of Trials until a First Success 3 Flipping a coin n times. Sample space: set of all sequences of n H ’s and T ’s. What Are Random Variables? A random variable for an experiment with sample space S is a function that assigns a number to each element of S . Example Random variable “number of heads” takes a sequence and tells us how many heads are in that sequence. Example: X ( HTHHT ) = 3 . X ( THTHT ) = 2 . 4 Example 2: Rolling two dice Random variable is “sum of the values showing on top of dice” . X ( ) = 5 X ( ) = 10 5 Random Variables • What Are Random Variables? • Binomial Probabilities • Expected Values • Expected Values of Sums and Numerical Multiples • Indicator Random Variables • The Number of Trials until a First Success 6 Bernoulli Random Variables A test in which the outcome is either a success or failure . Flipping a coin A head Examples: Answer to an exam question A correct answer A Drug trial A successful treatment Sucesss If such a test has P ( Success ) = p and P ( Failure ) = q = 1- p It is called a Bernoulli trial or Benoulli Random Variable with success probability p 7 Jakob Bernoulli b. 1654, d. 1705 Swiss Mathematician and Scientist. Famous for his work on probability the- ory (where Bernoulli trials come from) and calculus. He often collaborated with his brother Johann Bernoulli, another famous mathematician For more information, please see http://en.wikipedia.org/wiki/James Bernoulli 8 Flipping a coin # of heads. We are given an Independent trials process with two outcomes at each stage: success and failure . Examples: Student performance on a test # of correct answers Drug trials # of successful treatments We analyze: probability of exactly k successes in n independent trials with probability p of success on each trial. Such an independent trials process is called a Bernoulli trials process Quantity of Interest Note that this is the sum of Bernoulli Random Variables 9 Suppose we have 5 Bernoulli trials, with probability p success on each trial. What is the probability of By Independence , probability of a sequence of outcomes is product of probabilities of individual outcomes. So, probability of any sequence of 3 successes and 2 failures is p 3 (1- p ) 2 . More generally, in n Bernoulli trials, probability of a given sequence of k successes and n- k failures is (a) Success on first 3 trials and failure on last 2?...
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L17_RandomVariables_print - 1 COMP170 Discrete Mathematical...

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