Useful Distributions
Chuan Shi
Suppose we have a population X ∼ N(µ, σ 2 ), and x1 , x2 , · · ·, xn is a n size sample of the
population. The mean and variance of the sample are given as:
x̄ =
n
n
1�
1 �
xi and s2 =
(xi − x̄)2
n i=1
n − 1 i=1
(1)
The following distributions constructed with x̄ and s2 (or s) are very useful in Statistics. They
could be helpful for your homework.
�
σ2
x¯ ∼ N µ,
n
�
or
x̄ − µ
σ ∼ N(0, 1)
√
n
x̄ − µ
s ∼ t(n − 1)
√
n
(2)
(3)
(n − 1)s2
∼ χ2 (n − 1)
(4)
2
σ
Suppose there are two populations X ∼ N(µ1 , σ12 ) and Y ∼ N(µ2 , σ22 ), and a n1 size sample of
X, and a n2 size sample of Y . Means of the two samples are x̄ and ȳ, respectively. Variances of the
two samples are s21 and s22 , respectively. Then,
(x̄ − µ1 ) − (ȳ − µ2 )
�
σ12 σ22
+
n1 n2
(x̄ − µ1 ) − (ȳ − µ2 )
�
s2
s2
+
n1 n2
where
s2 =
∼ N(0, 1)
(5)
∼ t(n1 + n2 − 2)
(6)
(n1 − 1)s21 + (n2 − 1)s22
n1 + n2 − 2
The distribution (6) above can be used in the Hypothesis Test of means of two populations, in case
that σ1 and σ2 are unknown, but can be assumed equal.
�
�
s21
σ2
� 1 � ∼ F (n1 − 1, n2 − 1)
s22
σ22
1
(7)
MIT OpenCourseWare
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2.854 / 2.853 Introduction to Manufacturing Systems
Fall 2010
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