Between two distinct methods for manufacturing certain goods, the quality of goods produced by method i is a continuous random variable having distribution F i , i = 1 , 2 . Suppose that n goods are produced by method 2 and m by method 2. Rank the n + m goods according to quality, and let X j = { 1 if the i th best was produced from method 1 2 otherwise X 1 , X 2 , ... X n + m , which consists of n, l’s and m 2’s, let R denote the number of runs of 1. For instance, if n = 5 , m = 2 , and X = 1 , 2 , 1 , 1 , 1 , 1 , 2 ,then R = 2 . lf F 1 = F 2 (that is. if the two methods produce identically distributed goods), what are the mean and variance of R?
Between two distinct methods for manufacturing certain goods, the quality of goods produced by method i is a continuous random variable having distribution F i , i = 1 , 2 . Suppose that n goods are produced by method 2 and m by method 2. Rank the n + m goods according to quality, and let X j = { 1 if the i th best was produced from method 1 2 otherwise X 1 , X 2 , ... X n + m , which consists of n, l’s and m 2’s, let R denote the number of runs of 1. For instance, if n = 5 , m = 2 , and X = 1 , 2 , 1 , 1 , 1 , 1 , 2 ,then R = 2 . lf F 1 = F 2 (that is. if the two methods produce identically distributed goods), what are the mean and variance of R?
Solution Summary: The author explains how to find the mean and variance of R. The indicator random variable indicates if the best product has been made by method 1.
Between two distinct methods for manufacturing certain goods, the quality of goods produced by method i is a continuous random variable having distribution
F
i
,
i
=
1
,
2
. Suppose that n goods are produced by method 2 and m by method 2. Rank the
n
+
m
goods according to quality, and let
X
j
=
{
1
if the i th best was produced from method 1
2
otherwise
X
1
,
X
2
,
...
X
n
+
m
, which consists of n, l’s and m 2’s, let R denote the number of runs of 1. For instance, if
n
=
5
,
m
=
2
, and
X
=
1
,
2
,
1
,
1
,
1
,
1
,
2
,then
R
=
2
. lf
F
1
=
F
2
(that is. if the two methods produce identically distributed goods), what are the mean and variance of R?
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
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