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Unformatted text preview: AGENDA Probability Distributions Random variable concepts Bernoulli trials Binomial distribution RANDOM VARIABLES CONCEPTS Random variables A function that assigns a value to a possible outcome Discrete take on specific values in a range Continuous take on any values in a range Probability distribution Random variables values can be observed List of possible values and probability of observing each Cumulative Distribution Function Based on the probability distribution Probability a random variable is less than or = x PROBABILITY DISTRIBUTION AKA probability density List of possible values and probability f(x)=(usually some equation with input values) f(x)s sum to 1.0 f(1)=0.26, f(2)=0.50, f(3)=0.22, f(4)=0.02 CUMULATIVE DISTRIBUTION FUNCTION AKA Distribution Function Probability a random variable is less than or = x F(x)= F(1)=0.26, F(2)=0.76, F(3)=0.98, F(4)=1.00 Some cumulative distribution function values are tabulated in the back of the course textbook BERNOULLI TRIAL Bernoulli trials have only two possible outcomes for each trial Success = p Failure = 1-p P(success)=0.9, P(failure)=0.1 The probability for success or failure is the same percentage for each Bernouilli trial for all the trials BINOMIAL DISTRIBUTION Consists of a series of Bernouilli trials Outcomes of different Bernouilli trials are independent Prior outcomes have no impact on future outcomes Fixed number of Bernouilli trials, n For a single trial Probability of success is p Probability of failure is 1-p Use multiplication rule of probability for the trials BINOMIAL DISTRIBUTION Example Computer locking up while surfing the internet Probability of a reboot required is 0.1 for a half hour session BINOMIAL DISTRIBUTION Success = no reboot required= 0.9 Failure = reboot required = 0.1 For 3 different 30 minute sessions No successes, x=0, = FFF One success, x=1, = FFS, FSF, SFF Two successes, x=2, = FSS, SSF, SSF Three successes, x=3, = SSS BINOMIAL DISTRIBUTION Probabilities X=0, 0 successes and 3 failures, 1 way of 0.1 x 0.1 x 0.1 X=1, 1 success and 2 failures, 3 ways of 0.9 x 0.1 x 0.1 X=2, 2 successes and 1 failure, 3 ways of 0.9 x 0.9 x 0.1 X=3, 3 successes and 0 failures, 1 way of 0.9 x 0.9 x 0.9 Same as combinations (order does not matter) X=0, combination n=3 and r= 0 ways of (0.9^0 x 0.1^3) X=1, combination n=3 and r=1 ways of (0.9^1 x 0.1^2) X=2, combination n=3 and r=2 ways of (0.9^2 x 0.1^1) X=3, combination n=3 and r=3 ways of (0.9^3 x 0.1^0) We can devise a formula for this replacing r with x BINOMIAL PROBABILITY DISTRIBUTION Probability of observing x successes in n trials Input parameters x = number of success n = number of trials p = probability of success Have BINOMIAL CUMULATIVE DISTRIBUTION FUNCTION Total probability less than or equal to getting some x Input parameters x = number of success n = number of trials p = probability of success ENGINEERING EXAMPLE Marketing claim is that 60% of solar-heat installations result in a utility bill reduced by at least one third Probability that this happens in exactly 4 of 5 installations ENGINEERING EXAMPLE Marketing claim is that 60% of solar-heat installations result in a utility bill reduced by at least one third Probability that this happens in at least 4 of 5 installations ANOTHER ENGINEERING EXAMPLE USING TABLE 1 Probability that a column will fail is 0.05, among 16 columns What is the probability that at most two will fail? Means that 1 or 2 will fail B(2;16,0.05) Look in table 1 =0.9571 ANOTHER ENGINEERING EXAMPLE USING TABLE 1 Probability that a column will fail is 0.05, among 16 columns What is the probability that at least 4 will fail? Means 4 or 5, .16 will fail Can make use of the 1-p property of the CDF Look in table 1 1-B(3;16,0.05) 1-.9930=0.0070 LAST ENGINEERING EXAMPLE USING TABLE 1 Probability that 0.2 that any one student will dislike engineering statistics. What is the probability that 5 of 18 students randomly selected students dislike engineering statistics Table 1 b(5:18,0.20) B(5;18,0.20)-B(4;18,0.20) 0.8671-0.7164 0.1507 ...
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This note was uploaded on 11/21/2010 for the course INDE INDE2333 taught by Professor Chung during the Fall '09 term at University of Houston.
- Fall '09