Question

bectively. The ianctioe L) t a decreao the maximum occurs at the smallest val and is 0 otherwise sketch the that θ can seeune, eg the 1.1 Histogram Estimates of togram pmfs and pdfs . be a random sample on a random varial Let X,X be a n variable X with eds Flx we briefly discuss a histogram of the F. p/x), or the pdf, f(), of X depending on whe Oter than X being a discrete or cont. arametric form of the distribution as we did for the above discue likelihood estimates; hence, the histogram that she mf, ), or the pdf, (x), of X depending anple, which is an la) I whh discrete do ont nptions on the form of the distribution of X. particular. e nase assume a parametric form of tibution of fadon v estimator. We discuss the discrete situation first. we present ls often called The Distribution of X Is Discrete (PA a discrete random variable with pmf p(z). Suppoese first that the Assume that X is a discrete random Assume trois finite, say, D = {a1, . . . ,a,n). relative frequency of sample observations, which are equal to amator ofp %) is the define the statistics An intuitiveestimator which are equal to a,. For j space o Then the intuitive estimate of p(aj) can be expressed by the average 1.9) TL Thus the estimates {p 01 , p(0m)) constitutethe nonparametric estimate of the pmf p(x). Note thatl,(X)has a Bernoulli distribution with probability of success pla,). As Exercise 1.6 shows, our estimator of the pmf is unbiased. Suppose next that the space of X is infinite, say, D-fa we select a value, say, am, and make the groupings , a2 .. . In practice, et pam+) be the proportion of sample items that are greater than or equal to am+1. Then the estimates Plar),.m),lamtl)! form our estimate of 211

With using (1.9), What is the MGF(moment generating function) of this? Would you please solve this problem in detail?

Answer 1