Let X be the sample mean of a random sample of size 25 from a gamma distribution with density: 1 f(x) = x³e-1/3 x > 0. 6,30 Use the central limit theorem to find an approximate .954 confidence interval for μ= 43, the mean of this distribution.
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- Let X1, · · · , Xn be drawn independently from a distribution with density 1. f(x|θ) = 2/β, θ − 0.25β < x < θ + 0.25β, where β is a positive constant. 2. f(x|θ) = 2x/θ 2 , 0 < x < θ, θ > 0. 3. f(x|θ) = e θ−x for x > θ. For each of the above three cases, construct a 95% confidence interval for θ.Let Yn denote the nth order statistic of a random sample of size n froma distribution of the continuous type. Find the smallest value of n for which theinequality P(ξ0.9 < Yn) ≥ 0.75 is true.Let X1, X2,...Xn be a random sample of size n from a normal distribution with mean u and variance o2. Let Xn denote the sample average, defined in the usual way. PROVE E [Xn] = u
- Find the probability that the range of a random sample of size 4 from theuniform distribution having the pdf f(x) = 1, 0 < x < 1, zero elsewhere, is lessthan 12 .Let Y1 < Y2 < · · · < Yn be the order statistics of a random sample of size nfrom a distribution with pdf f(x) = 1, 0 < x < 1, zero elsewhere. Show that thekth order statistic Yk has a beta pdf with parameters α = k and β = n − k + 1.If the probability density of X is given by f(x) =kx3(1 + 2x)6 for x > 00 elsewhere where k is an appropriate constant, find the probabilitydensity of the random variable Y = 2X 1 + 2X . Identify thedistribution of Y, and thus determine the value of k.
- Suppose that Y1,Y2,Y3 denote a random sample from an exponential distribution with density function f(y) = Consider the following four estimators of θ: ?1θe−y/θ, y>0,0, otherwise. θˆ =Y, θˆ =Y1+Y2, θˆ =Y1+2Y2, θˆ =Y1+Y2+Y3 =Y ̄. 11223343 Which estimators are unbiased? Among the unbiased estimators, which has the smallest variance?A continuous random variable X has a pdf of the form: f(x) = (824/9) x^2, for 0.01 < X < 0.32. Calculate the standard deviation (sigma) of X.Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with density curve f(x) = (1/3)e(-x/3) for x > 0 and f(x) = 0 otherwise. What is the probability that response time exceeds 5 seconds? What is the probability that response time exceeds 10 seconds?