概率统计英文 chapter Continuous Random Variables and Probability Distributions.ppt

The Exponential Distribution DEFINITION: X is said to have an exponential if the pdf of X is The exponential pdf is a special case of the general g-amma pdf Expression (4.7) in which and has been repaced by . The mean and varance of X are then * Unlike the general gamma pdf, the exponential pdf can be easily integrates. In particular, the cdf of X is x 2 1 .5 Figure 4.27 * Application of the Exponential Distribution PROPOSITION: Suppose that the number of events occurring in any time interval of length t has a Poiss-on distribution with parameter and that num-bers of occurrences in nonoverlapping intervals are inde-pendent of one another . Then the distribution of elapsed time between the occurrence of two succe-ssive events is exponential with parameter . Example 4.23 Suppose that calls are received at a 24-hour “suicide hotline” according to a Poisson process with rate call per day. Then the number of days X between successive calls has an exponential distribution with parameter values .5, so the probability tahat more tahan 2 days elapse between calls is The expected time between successive calls is 1/.5=2 days. * The parameter is called the number of degrees of free-dom of X . The symbol is often used in place of “chi-squared The Chi-squared distribution DEFINITION : Let be a positive integer . Then a random variable X is said to have a chi-aquared distribution with parameter if the pdf of X is the gamma density with and . The pdf of a chi-aquared rv is thus * 4.5 Other Continuous Distribution The Weibull Distribution DEFINITION: A random variable X is said to have a Weibull distribution with parameters and if the cdf of X is Integrating to obtain E(X) and E( ) yields * PROPOSITION : The cdf of a Weibull rv having parameters and is The Lognormal Distribution DETINITION : A nonnegative rv X is said to have a lognormal distribution if