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Stat 103 Test 2
Covers: Chapters 5, 6 (not 6.5), 7.1, 7.2, 8.1, 8.2, 8.3, 8.4, 11.1. 11.2, 15.1
Chapter 5: Two Random Variables
5.1 – Distributions
1.
Joint (bivariate) distribution of X and Y – definition
2.
Marginal distribution of X
a.
Used
when we are only interested in X, yet have to work with the joint distribution
of X and Y
b.
Row sums are calculated and placed in the margin, thus the name
i.
When p(x)p(y) agrees everywhere with the original table of p(x,y), X and Y are
independent
ii.
Whenever X and Y are independent, then the rows of the table p(x,y) will be
proportional and so will the columns
5.2 – A Function of Two Random Variables
5.3 – Covariance
3.
Covariance is used to measure how two variables X and Y vary together
a.
Can indicated whether X and Y have a positive, negative, or zero relation
b.
Covariance = sum of all the calculated variables
c.
When covariance = 0, X and Y are uncorrelated
i.
One way for this happens is if X and Y are independent
4.
Covariance depends on units in which X and Y are measured
a.
Correlation (P) eliminates this problem
i.
Correlation is the standardized version of covariance
ii.
Measures the degree of linear relation between X and Y
b.
Perfect
positive linear relation – P = 1, perfect negative linear relation – P = 1
5.4 – Linear Combination of Two Random Variables
5.
Linearity/Additivity property of the expectation operator
a.
E(aX + bY) = aE(X) + bE(Y)
b.
Linearity property allows us to simply use the individual means E(X) and E(Y),
instead of working through the whole bivariate table
6.
Variance
a.
Variance is defined by squared deviations
b.
We add variances, NOT standard deviations
Summary
7.
A pair of random variables X and Y has a joint probability distribution of p(x,y) from
which the distributions p(x) and p(y) can be fond in the margin
8.
X and Y can be called independent if p(x,y) = p(x)p(y) for all x and y
9.
Just as variance measures how much one variable varies, so covariance measures how
much two variables vary together
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View Full Document Chapter 6: Sampling
6.1 – Random Sampling
10. Basic deductive question in statistics: what can we expect of a random sample drawn
from a known population?
a.
How we
collect
data is at least as important as how
analyze
it
b.
A sample should be representative of the population
i.
Random sampling is often the best way to achieve this
ii.
If a sample is not random, it may be so biased that it is worse than useless
iii.
Quantity alone can be deceptive  a large biased sample may look good because
of its size but it may just contain the same bias being repeated over and over
11. In statistics, population = the total collection of objects or people to be studied, from
which a sample is to be drawn
12. Each individual observation in a random sample has the population probability
distribution p(x)
a.
From p(x), we calculate the population mean and standard deviation, which are also
the mean and standard deviation of an individual observation
13. A sample is called a simple random sample (SRS) if each individual in the population is
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This document was uploaded on 06/28/2011.
 Spring '09

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