概率与统计英文chapter Joint Probability Distributions and Random Samples.ppt

DEFINITON The rv’s X1,X2,…,Xn are said to form a (simple) random sample of size n if The Xi’s are independent rv’s. 2, Every Xi has the same probability distribution. Deriving the Sampling Distribution of a Statistic Probability rules can be used to obtain the distribution of a statistic provided that it is a “fairly simple” function of the Xi’s and either there are relatively few different X values in the population or else the population distribution has a “nice”form.Our next two examples illustrate such situations. Example 5.21 The time that it takes to serve a customer at the cash register in a minimarket is a random variable having an exponential distribution with parameter λ.Suppose are service times for two different customers,assumed independent of each other.Consider the total service * time To=X1+X2 for the two customers,also a statistic.The cdf of To is,for t≥0, The region of integration is pictured in Figure 5.9. x1+x2=t (x1,t-x1) x1 x2 Figure 5.9 Region of integration to obtain cdf of T0 in Figure 5.21 The pdf of T0 is obtained by differentiating FT0(t); * This is a gamma pdf (α=2 and β=1/λ).The pdf of X=T0/2 is obtained from the relation {X≤x}iff{T0 ≤2x} as Simulation Experiments The second method of obtaining information about a statistic’s sampling distribution is to perform a simulation experiment.This method is usually used when a derivation via probability rules is too difficult or complicated to be carried out.Such an experiment is virtually always done with the aid of a computer.The following characteristics of an experiment must be specified: The statistic of interest ( , S,a particular trimmed mean,etc.) 2. The population distribution (normal with μ=100 and σ=15,uniform with lower limit A=5 and upper limit B=10,etc) * 3. The sample size n (e.g.,n=10 or n=50) 4. The number of replication k (e.g.,k=500) Then use a computer to obtain k different random samples,each of size n ,from the designated population distribution.For each such sam