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Suppose that Y1, Y2, Y3 denote a random sample from an exponential distribution with density
Consider the following live estimators of θ:
- a. Which of these estimators are unbiased?
- b. Among the unbiased estimators, which has the smallest variance?
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Chapter 8 Solutions
Mathematical Statistics with Applications
- For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)arrow_forwardSuppose random variable X has a density function f ( x ) = { 2 /x 2 , 1 ≤ x ≤ 2 0 , o t h e r w i s e . Then E[X4] =?arrow_forward6.) Suppose X is continuously uniformly distributed on [−2, 2]. Let Y = X2. What is the density function of Y? What is the expected value of Y?arrow_forward
- Suppose X and Y are independent and identically distributed (i.i.d.) randomvariables, each with the uniform distribution on [0, 1]. What is the cumulative distributionfunction and the density function of XY ?arrow_forwardLet X and Y be random variables with the joint density function f(x,y)=x+y, if x,y element of [0,1], and f(x,y)=0,elsewhere. Find the expected value of the random variable Z = 10X+14Y.arrow_forwardLet x be a continuous random variable with the density function: f(x) = 3e-3x when x>0 and 0 else Find the variance of the random variable x.arrow_forward
- 1) Let X1, X2, ..., Xn be a sample of n units from a population with a probability density function f (x I θ)=θxθ-1 , 0<x<1, θ>0 . According to this: Find the estimator of moments for the parameter θ.arrow_forwardSuppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise c=1/2 P(X < 1), Determine whether X and Y are independentarrow_forwardSuppose that the random variables X and Y have a joint density function f(x,y).prove that Cov(X,Y)=0 if E(X|Y=y) does not depend on yarrow_forward
- 5.)Suppose X is continuously uniformly distributed on [1, 4]. Let Y = ln(X). What is the density function for Y ? (Include the bounds for Y .)arrow_forward1) Let x be a uniform random variable in the interval (0, 1). Calculate the density function of probability of the random variable y where y = − ln x.arrow_forwardSuppose that the random variables X and Y have a joint density function given by: f(x,y)={cxy for 0≤x≤2 and 0≤y≤x, 0 otherwise Find the constant c, P(Y≥1/2), P(X < 2, Y >1/2), P(X < 1), Determine whether X and Y are independent.arrow_forward
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