What is the purpose of principal component analysis? Given the covariance matrix of order 2× ,2 explain how would you extract the principal components. Also, explain how would you find the proportion of total population variance for all the principal components.
Principal Component Analysis (PCA) is a statistical technique primarily used for dimensionality reduction while preserving as much variability as possible in the dataset. The key purpose of PCA is to identify patterns in high-dimensional data and express the data in such a way that it highlights their similarities and differences. It achieves this by transforming the original correlated features into a set of new uncorrelated features known as principal components. These new features are ordered by their variance, with the first principal component explaining the highest variance, the second explaining the next highest, _____ _________ ______ ________ __________ ___ _____ ______ _________ _____.
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The joint distribution of the random variables X and Y is given by:
| y/x | -1 | 0 | 1 |
| -1 | α | β | α |
| 0 | β | α | β |
| 1 | α | β | α |
In an investigation related to a specific type of scores of men and women aged 65 to 70, the mean verbal and performance scores for 101 subjects were found to be:
The sample covariance matrix of the scores was
In order to test the null hypothesis that observations came from a population with mean vector apply a suitable test statistic. You may consider α = 0.01 for the test.
In a certain state, 58 landfills are classified according to their concentration of three hazardous chemicals: arsenic, barium and mercury. Suppose that the concentration of each one of the three chemicals is characterized as either high or low. If a landfill is chosen at random from among 58 landfills, given the following configuration:
| Barium | ||||
| High Mercury | Low Mercury | |||
| Arsenic | High | Low | High | Low |
| High | 1 | 3 | 5 | 9 |
| Low | 4 | 8 | 10 | 18 |
The joint distribution of the random variables X and Y is given by:
| x /y | −1 | 0 | 1 |
| −1 | α | β | α |
| 0 | β | 0 | β |
| 1 | α | β | α |
| 1 | 2 | 3 | 4 | |
| 1 | 0 | 0 | 1 | 0 |
| 2 | 1 | 0 | 0 | 0 |
| 3 | 1/2 | 1/2 | 0 | 0 |
| 4 | 1/3 | 1/3 | 1/3 | 0 |
i) Find the probabilit
b) It is claimed that the functionis the joint distribution function of the random variables X and Y. Then:
ii) calculate the probability
A study about a population showed that the mobility of population of a state to a village, town and city is in the following percentages:
| From | To | ||
| Village | Town | City | |
| Village | 60 | 25 | 15 |
| Town | 10 | 70 | 20 |
| City | 10 | 30 | 60 |
Consider the following system considering of two switches I and II between two points A and B:
A signal is sent from the point A to point B and is received at B if both the switches I and II are closed. It is assumed that the probabilities of I and II being closed are 0.8 and 0.6 respectively and that P [II is closed | I is closed] = P [II is closed].
Find:
i) The probability that signal is received at B.
ii) The conditional probability that switch I was open, given that the signal was not received at B,
iii) The conditional probability that switch II was open, given that the signal was not received at B.
A community has two police cars, which operate independently of one another. The probability that a specific car will be available when needed is 0.99.
i) What is the probability that neither car is available when needed?
ii) What is the probability that a car is available when needed?