Statistical Sciences 2141A/B Lecture Notes - Lecture 2: Binomial Distribution, Project Echo, V Speeds
17 May 2016
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Soln: respectively given by: is a random variable with mean. Let (cid:2869), , be n independent random variables with respective means (cid:2869) and variances (cid:2869)(cid:2870) (cid:2870). Approximate 95% confidence intervals for a proportion p and a mean are. 2. 5-1) let the independent r. v. s (cid:2869) and (cid:2870) have parameters (cid:2869)=1, (cid:2870)=2, (cid:2869)(cid:2870)=4, (cid:2870)(cid:2870)=9, respectively. Find the mean and the variance of y=3(cid:2869)-2(cid:2870). Mean(y) = e(3(cid:2869) 2(cid:2870)(cid:4667)=3e((cid:2869)) 2e((cid:2870)) = 3*1-2*2=-1 var(y) = var(3(cid:2869) 2(cid:2870)) = 9var((cid:2869))+4var((cid:2870))=9*4+4*9=72. 2. 5-2) let (cid:2869) and (cid:2870) be independent r. v. s with respective variances (cid:2869)(cid:2870)=k, (cid:2870)(cid:2870)=2. Given that the variance of y = 3(cid:2870) (cid:2869) is 25, find k. Var(y) = var(3(cid:2870) (cid:2869)(cid:4667) = 9var((cid:2870))+var((cid:2869)) = 25 k=var((cid:2869)(cid:4667) = 7. 2. 5-3) let be the mean of a random sample of size 9 from a distribution with p. m. f. f(x) = x/6, x=1,2,3. Find the mean and the variance of . This variance is obtained from the variance formula in chapter2. Soln: (a) since we know that the r. v. s are independent:
