logit distribution 逻辑分布.pdf

Virtual Laboratories 4. Special Distributio ns 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 12. The Logistic Distribution The logistic distribution has been used for various growth models, and is used in a certain type of regression, known appropriately as logistic regression . The Standard Logistic Distribution 1. Show that the function F given below is a distribution function. x e F ( x ) = ,鈥儀 鈭 鈩 1 e x The distribution defined by the function in Exercise 1 is called the (standard) logistic distribution . 2. Suppose that X has the logistic distribution. Find 鈩?鈭? X 2). 3. Show that the probability density function f of the logistic distribution is given by x e f ( x ) = ,鈥儀 鈭 鈩 (1 e x )2 4. Sketch the graph of the logistic density function f . In particular, show that a. f is symmetric about x = 0. b. f is increasing on (鈭掆垶, ) and decreasing on 0 (0, 鈭?. Thus, the mode occurs at x = 0 5. In the random variable experiment, select the logistic distribution. Note the shape and location of the density function. Run the simulation 1000 times with an update frequency of 10 and note the apparent convergence of the empirical density to the true density. 6. Show that the quantile function is 鈭? p