There is only one parameter , and by method of moments we know the sample mean is equal to the expectation of the random variable X.
Here sample mean is
Also
So equating the two gives
Thus the point estimator is
2. Let X=(X,, X2, ,X") be a random sample with size n taken from population has...
Ex (5) Let X = (Xi, X2, ,X") be a random sample with size n taken from population has e-부) a) 71 2 X is an unbiased estimator of τ (θ)-2(J+ b) T-X is a consistent estimator of τ (9) (J+ β fx(x ; θ) , β < x <。。.Show that 2)
7-27. Let X1, X2,..., X, be a random sample of size n from a population with mean u and variance o?. (a) Show that X² is a biased estimator for u?. (b) Find the amount of bias in this estimator. c) What happens to the bias as the sample size n increases?
Let X1, X2, ..., Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) max(X1,X2, ...,Xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for e.
(1 point) Let X1 and X2 be a random sample of size n= 2 from the exponential distribution with p.d.f. f(x) = 4e - 4x 0 < x < 0. Find the following: a) P(0.5 < X1 < 1.1,0.3 < X2 < 1.7) = b) E(X1(X2 – 0.5)2) =
Let X1, X2, ...,Xn be a random sample of size n from a population that can be modeled by the following probability model: axa-1 fx(x) = 0 < x < 0, a > 0 θα a) Find the probability density function of X(n) = max(X1, X2, ...,xn). b) Is X(n) an unbiased estimator for e? If not, suggest a function of X(n) that is an unbiased estimator for 0.
Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3. ) Let Zn 3 n n-2ућ. Find Mz, (t), the mgf of Žn. Then find the limiting mgf limn→oo MZm (t). What is the limiting distribution of Z,'? Let X1, X2, .. , Xn be a random sample of size n from a geometric distribution with pmf =0.75 . 0.25z-1, f(x) X-1.2.3. ) Let Zn 3...
4.20 $ 4.19. Let X, X2,..., X2, be a random sample of size n = 20 from an Nou, 2) population. Specify each of the following completely. (a) The distribution of (X -M -'(X -M) (b) The distributions of X and Vn(X - M) (c) The distribution of (n - 1) S 4.20. For the random variables X1, X2,..., X20 in Exercise 4.19, specify the distribution of B(198)B' in each case. (a) B - 1 - - 0 0 0]...
A random sample of size n = 21, taken from a normal population with a standard deviation 04 =5, has a mean X4 = 90. A second random sample of size n2 = 37, taken from a different normal population with a standard deviation o2 = 4, has a mean X2 = 39. Find a 94% confidence interval for 11 - H2 Click here to view page 1 of the standard normal distribution table. Click here to view page 2...
Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = (1/θ)e^(−x/θ) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ. Let X1, X2, ... , Xn be a random sample of size n from the exponential distribution whose pdf is f(x; θ) = θe^(−θx) , 0 < x < ∞, 0 <θ< ∞. Find the MVUE for θ.
4.19. Let Xl, X2, ... , X20 be a random sample of size n = 20 from an N(u, E) population. Specify each of the following completely (a) The distribution of (X (b) The distributions of X and Vn(X- ) 'E(X-H) (c) The distribution of (n - 1) S etibuti. 4.19. Let Xl, X2, ... , X20 be a random sample of size n = 20 from an N(u, E) population. Specify each of the following completely (a) The distribution...