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 + ...
Calculating probability for the Standard Normal Curve
1.
Assume the mean is zero, the standard deviation is one, and it is associated with the distribution of z values.
Each problem is worth 2 points, 1 point for drawing out the curve and shading the area requested and 1 point
for the answer.
a. What is the P(z > 0)?
b. What is the P(z < 1.0)?
C. What is the P(z <-1.0)?
Starting with the finished version of Example 6.2, attached, change the decision criterion to "maximize expected utility," using an exponential utility function with risk tolerance $5,000,000. Display certainty equivalents on the tree.
a. Keep doubling the risk tolerance until the company's best strategy is the same as with the EMV criterion—continue with development and then market if successful.
The risk tolerance must reach $ 160,000,000 before the risk averse company acts the same as the EMV-maximizing company.
b. With a risk tolerance of $320,000,000, the company views the optimal strategy as equivalent to receiving a sure $____________ , even though the EMV from the original strategy (with no risk tolerance) is $ 59,200.
Starting with the finished version of Example 6.2, attached, change the decision criterion to "maximize expected utility," using an exponential utility function with risk tolerance $5,000,000. Display certainty equivalents on the tree.
a. Keep doubling the risk tolerance until the company's best strategy is the same as with the EMV criterion—continue with development and then market if successful.
The risk tolerance must reach $ ____________ before the risk averse company acts the same as the EMV-maximizing company.
b. With a risk tolerance of $320,000,000, the company views the optimal strategy as equivalent to receiving a sure $____________ , even though the EMV from the original strategy (with no risk tolerance) is $ ___________ .
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