X follows a normal distribution with a mean of 6 and a standard deviation of 2 Find P 5 x 8 Variables in an experiment sometimes have relation x exp w If w itself follows a Gaussian Distribution with a mean q and variance w2 then x is a log normal random variable with a PDF ln x q 2 f x exp 2 2w xw 2 1 mean variance w2 q 2 2q w 2 e e e 2 w2 1 x exp w w itself follows a Gaussian Distribution with a mean q and variance w2 How can we find P x 1000 P x 1000 P ln x ln 1000 P w 6 907 w has a Normal Distribution and given the mean and variance we can figure this out using the z table Often used to model time until failure f x x 1 x exp Shape parameter Scale parameter Can get a wide variety of shapes predicting different behavior ceramics Used to model the strength of brittle materials glasses Only discrete values possible Example building has 2000 phone lines let X be the number of phone lines in use at a particular time Discrete random variable Distribution often described as a list of values along with probabilities P X 0 0 6561 P X 1 0 2916 P X 2 0 0486 P X 3 0 0036 P X 4 0 0001 For a discrete random variable X with possible values x1 x2 x3 Probability mass function f xi P X xi n where f x 1 i 1 i n Mean xi f xi P X 0 0 6561 P X 1 0 2916 P X 2 0 0486 P X 3 0 0036 P X 4 0 0001 i 1 n 2 x f xi Variance i 2 i 1 n repeated trials each trial results in either success or failure Probability of success p Called a binomial experiment follows a binomial distribution Flip a coin 3 times what is the probability of getting 2 heads n x PMF f x p 1 p n x x x 0 1 n n n x x n x Probability of success Total number of trials Number of trials w success 3 2 3 1 3 Back to the coin f 2 0 5 0 5 3 0 5 8 2 Mean np Variance 2 np 1 p n x n x f x p 1 p x x 0 1 n n n x x n x Answer 0 2326 0 3721
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