Suppose that n = 5 observations are taken from the uniform pdf , f Y ( y ; θ ) = 1 / θ , 0 ≤ y ≤ θ , where θ is unknown. Two unbiased estimators for θ are θ ^ 1 = 6 5 ⋅ Y max and θ ^ 2 = 6 ⋅ Y min Which estimator would be better to use? ( Hint: What must be true of Var ( Y max ) and Var ( Y min ) given that f Y ( y ; θ ) is symmetric?) Does your answer as to which estimator is better make sense on intuitive grounds? Explain.
Suppose that n = 5 observations are taken from the uniform pdf , f Y ( y ; θ ) = 1 / θ , 0 ≤ y ≤ θ , where θ is unknown. Two unbiased estimators for θ are θ ^ 1 = 6 5 ⋅ Y max and θ ^ 2 = 6 ⋅ Y min Which estimator would be better to use? ( Hint: What must be true of Var ( Y max ) and Var ( Y min ) given that f Y ( y ; θ ) is symmetric?) Does your answer as to which estimator is better make sense on intuitive grounds? Explain.
Solution Summary: The author explains that stackreltheta_1 makes sense on intuitive grounds.
Suppose that
n
=
5
observations are taken from the uniform pdf,
f
Y
(
y
;
θ
)
=
1
/
θ
,
0
≤
y
≤
θ
, where
θ
is unknown. Two unbiased estimators for
θ
are
θ
^
1
=
6
5
⋅
Y
max
and
θ
^
2
=
6
⋅
Y
min
Which estimator would be better to use? (Hint: What must be true of
Var
(
Y
max
)
and
Var
(
Y
min
)
given that
f
Y
(
y
;
θ
)
is symmetric?) Does your answer as to which estimator is better make sense on intuitive grounds? Explain.
Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.
Suppose that a person’s score X on a mathematics aptitude test is a number between 0 and 1 and that his score Y on a music aptitude test is also a number between 0 and 1. Suppose further that in the population of all college students in Belize, the scores X and Y redistributed according to the following joint pdf.
f(x,y) = 2/5(2x + 3y) for 0≤x≤1, 0≤y≤1 and 0 otherwise
What proportion of college students obtain a score greater than 0.8 on the mathematics test?
If a student’s score on the music test is 0.3, what is the probability that his score on the mathematics test will be greater than 0.8?
If f(x, y) = 1/(xy), then the marginal pdf of X is (log x)/y. Is this statement True/False?
Plot f (x) = ln x − 5 sin x on [0.1, 2] and approximate both the critical points and the extreme values.
Chapter 5 Solutions
An Introduction to Mathematical Statistics and Its Applications (6th Edition)
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