If Y 1 , Y 2 , ... , Y n are random observations from a uniform pdf over [ 0 , θ ] , both θ ^ 1 = ( n + 1 n ) ⋅ Y max and θ ^ 2 = ( n + 1 ) ⋅ Y min are unbiased estimators for θ . Show that Var ( θ ^ 2 ) /Var ( θ ^ 1 ) = n 2 .
If Y 1 , Y 2 , ... , Y n are random observations from a uniform pdf over [ 0 , θ ] , both θ ^ 1 = ( n + 1 n ) ⋅ Y max and θ ^ 2 = ( n + 1 ) ⋅ Y min are unbiased estimators for θ . Show that Var ( θ ^ 2 ) /Var ( θ ^ 1 ) = n 2 .
Solution Summary: The author explains that if Var(stackreltheta _2) is a random sample of continuous random variables, the probability density function of the largest order statistic
If
Y
1
,
Y
2
,
...
,
Y
n
are random observations from a uniform pdf over
[
0
,
θ
]
, both
θ
^
1
=
(
n
+
1
n
)
⋅
Y
max
and
θ
^
2
=
(
n
+
1
)
⋅
Y
min
are unbiased estimators for
θ
. Show that
Var
(
θ
^
2
)
/Var
(
θ
^
1
)
=
n
2
.
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.
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