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MATH 4030 – 6A MORE DISTRIBUTIONS FOR CONTINUOUS RV’S 1 • Uniform • Log-Normal • Gamma & Exp • Beta • Weibull UNIFORM U(,) (SEC. 5.5): f ( x) C, x . • Value of C: Axiom of probability • Graphs of f(x) and F(x) • Probability • Mean E[X] and variance Var[X] LOG-NORMAL DISTRIBUTION (SEC. 5.6): Pdf: ln x 1 1 2 2 , for x 0, 0, f ( x) 2 x e 0, otherwise . 2 3 Parameters: and > 0 LOG-NORMAL AND NORMAL: When X has a Log-Normal distribution, then ln(X) has a normal distribution with the same parameters. X ~ LN , ln X ~ N , 2 Mean: X e Variance: e 2 X 2 2 e ln X , ln2 X 2 . 2 2 2 . ln( X ) ln X ! 1 . 4 Probability: P(a X b) P(ln a ln X ln b) GAMMA DISTRIBUTION GAMMA(,) (SEC. 5.7): Parameters: > 0 and > 0 Pdf: x 1 1 x e , for x 0, 0, 0, f ( x) 0, otherwise . where () is a value of the gamma function: x 1e x dx 5 0 MEAN AND VARIANCE: . . 2 2 Used to model a continuous random variable that takes positive values and has asymmetric distribution. (rainfall, energy consumption, survival time,…) Represent a large family of continuous distributions with its flexibility of two parameters. (chi-square, exponential, …) 6 EXPONENTIAL DISTRIBUTION: A SPECIAL CASE OF GAMMA DISTRIBUTION Exp( ) Gamma(1, ) Pdf: 1 x e , f ( x) 0, for x 0, 0, otherwise . Mean: Variance: . 2 2 7 . EXPONENTIAL V.S. POISSON continuous discrete Assuming 3 arrivals per hour on average. • Let X be the number of arrivals in an hour, then X has Poisson distribution with parameter = 3. 8 • Let Y be the time observed between two arrivals. Then Y have exponential distribution with parameter = 1/3. Beta Distribution Beta(,) (Sec. 5.8): Pdf: x 1 1 x 1 1 1 x 1 x , for 0 x 1, 0, 0, f ( x ) B , 0, otherwise . Parameters: > 0 and > 0 Variance: . . 2 1 2 9 Mean: 10 Graph of Beta Distribution Beta(,): The Weibull Distribution (Sec. 5.9) Pdf: x e f ( x) 0, 1 x , for x 0, 0, 0, Parameters: > 0 and > 0. otherwise . x 1e x , f ( x) 0, 1 Let y x , x y , dx for x 0, 0, 0, otherwise . 1 1 y CDF is F ( x) 1 e 1 x dy , x 0. Probability can be easily calculated. Mean and Variance: x x 1 x e dx (Let y x ) 0 1 u 0 1 e du u 1 1 1 . 2 2 1 2 1 1 . 13 2 PROBABILITY DISTRIBUTIONS IN R Distribution Beta Binomial Chi-Square Exponential F Gamma Geometric pbeta pbinom pchisq pexp pf pgamma pgeom qbeta qbinom qchisq qexp qf qgamma qgeom dbeta dbinom dchisq dexp df dgamma dgeom rbeta rbinom rchisq rexp rf rgamma rgeom Hypergeometric phyper qhyper dhyper rhyper Log Normal Negative Binomial Normal Poisson Student t Uniform Weibull plnorm qlnorm dlnorm rlnorm pnbinom qnbinom dnbinom rnbinom pnorm ppois pt punif pweibull qnorm qpois qt qunif qweibull dnorm dpois dt dunif dweibull rnorm rpois rt runif rweibull Cumulative probability Functions Quantile of distribution Probability value or density value Randomly generated value from the distribution