Let x 1 , x 2 , ⋯ , x n be independent random variables, each with density function f ( x ) , expected value μ , and variance σ 2 . Define the sample mean by x ¯ = ∑ i = 1 n x i . Show that E ( x ¯ ) = μ , and Var ( x ¯ ) = σ 2 / n . (See Problems 5.9 , 5.13 , and 6.15. .
Let x 1 , x 2 , ⋯ , x n be independent random variables, each with density function f ( x ) , expected value μ , and variance σ 2 . Define the sample mean by x ¯ = ∑ i = 1 n x i . Show that E ( x ¯ ) = μ , and Var ( x ¯ ) = σ 2 / n . (See Problems 5.9 , 5.13 , and 6.15. .
Let
x
1
,
x
2
,
⋯
,
x
n
be independent random variables, each with density function
f
(
x
)
, expected value
μ
,
and variance
σ
2
.
Define the sample mean by
x
¯
=
∑
i
=
1
n
x
i
.
Show that
E
(
x
¯
)
=
μ
,
and
Var
(
x
¯
)
=
σ
2
/
n
.
(See Problems
5.9
,
5.13
,
and
6.15.
.
Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.
Let X be a continuous random variable with mean μ and standard deviation σ. If X is transformed to Y = 2X + 3, what are the mean and standard deviation of Y?
Let X be a random variable with mean E[X] = 20 and variance Var(X) = 3.
Define Y = 3 – 6X. Calculate the mean and variance of Y.
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