| 1.(20%) |
Let X1,X2,... be a sequence of random
variables that converges in probability to a constant a. Assume that P
(Xi>0)=1 for all i. |
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| a. |
Show that the sequence Y1,Y2,...,
defined by  ,
converges in probability to  . |
| b. |
Show that, if a > 0, the sequence Y1,
Y2,..., defined bye Yi = a /
Xi, converges in probability to 1. |
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| 2.(15%) |
Let XI, i =1, 2, 3, be independent with n(i,
i2) distributions. For each of the following situations,
use the XiS to construct a statistic with the indicated
distribution. |
| |
| a. |
Chi squared with 3 degrees of freedom. |
| b. |
t distribution with 2 degrees of freedom. |
| c. |
F distribution with 1 and 2 degrees of freedom. |
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| 3.(30%) |
For each of the following distributions let X1, ...,
Xn be a random sample. Find a minimal sufficient statistic
for θ. |
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| a. |
 , -∞ < x
<∞, -∞ <θ<∞ |
| b. |
 , θ< x <∞,
-∞ <θ< ∞ |
| c. |
 , -∞ < x
< ∞, -∞ <θ< ∞ |
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| 4.(10%) |
One observation is taken on a discrete random variable X with pmf f
(x|θ), where  . Find
the MLE of θ. |
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| 5.(15%) |
Let X1, ...,Xn be a random sample
from a gamma (α,β) population. |
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Find the MLE of β, assumingαis known. |
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| 6(10%) |
In 1,000 tosses of a coin, 560 heads and 440 tails appear. Is it reasonable
to assume that the coin is fair? Justify your answer. |
