The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study. Suppose that the joint p.m.f of X and Y is modeled as fxy (x, y) = c(x+y), x=1, 2, 3 and y=1, 2, 3
The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study. Suppose that the joint p.m.f of X and Y is modeled as fxy (x, y) = c(x+y), x=1, 2, 3 and y=1, 2, 3
The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study. Suppose that the joint p.m.f of X and Y is modeled as fxy (x, y) = c(x+y), x=1, 2, 3 and y=1, 2, 3
Exercise 4.1 The number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are under study.
Suppose that the joint p.m.f of X and Y is modeled as fxy (x, y) = c(x+y), x=1, 2, 3 and y=1, 2, 3 Determine the value of c.
Determine the marginal p.m.f of X. Find also the mean and variance of X.
Determine the conditional p.m.f of Y given X=2. Find also the conditional mean and conditional variance of Y given X=2.
Check if the number of defects on the front side (X) of a wooden panel and the number of defects on the rear side (Y) of the panel are independent.
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Transcribed Image Text:Exercise 4.1 The number of defects on the front side (X) of a wooden panel and the number of defects
on the rear side (Y) of the panel are under study.
a. Suppose that the joint p.m.f of X and Y is modeled as fry (x, y) = c(x+y), x=1, 2, 3 and y=1, 2, 3
Determine the value of c.
b. Determine the marginal p.m.f of X. Find also the mean and variance of X.
c. Determine the conditional p.m.f of Y given X=2. Find also the conditional mean and conditional
variance of Y given X=2.
d. Check if the number of defects on the front side (X) of a wooden panel and the number of defects on
the rear side (Y) of the panel are independent.
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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