1. Chapter 3 Properties of Random Variables
Moments and Expectation

2. Review Experiment Random Variable Observation Realization Parameter
Sample Statistic

3. Moments
One way to quantify the location and some measures of the shape of the pdf.

4. First moment about the origin

5. ith moment about the origin
continuous random variable
discrete random variable

6. ith central moment about the mean, m

7. Expected value of a random variable X
X continuous
X discrete

8. Expected value of function of X, g(X)
X continuous
X discrete

9. Expected value and the first moment about the origin
Comparing to
You can see that the expected value of the random value x is the first moment about the origin.

10. Rules for finding expected values

11. Measures of central tendency
Arithmetic mean
Geometric mean
Median
Mode
Weighted Mean

12. Mean, mx, or average value
Mean of a r.v. X is its expected value.
Sample estimate is the arithmetic average.
Arithmetic mean of grouped data (k is number of groups, n is total number of observations, ni is the number of observations in group i, xi is the class mark of the ith group.

13. Geometric mean
Used when the ratio of two consecutive observations is either constant or nearly constant.
The logarithm of the population geometric mean would be the expected value of the logarithm of X.

14. Median, Xmd
The observation such that half of the values in the sample lie on either side of Xmd. The median may not exist.
Population median, mmd would be the value satisfying:
X continuous
X discrete

15. Mode, mmo
Most frequently occurring value. The sample or population may have none, one or more than one mode.
Population mode is the value of X maximizing px(x).
X continuous
X discrete

16. Weighted mean
Used for describing the central tendancy of grouped data.

17. Measures of Dispersion
Measures of the spread of the data
Range
Variance

18. Range Difference between the largest and smallest sample values.
For a population this interval often ranges from - ∞ to ∞ or from 0 to ∞.
The sample range is a function of only 2 of the sample values, but does convey some idea of the spread of the data.
Disadvantage of range: does not reflect frequency or magnitude of values that deviate from the mean.
Occasionally use the relative range
Relative range =

19. Variance, s2 Defined as the second moment about the mean.
The average squared deviation from the mean. For a discrete population of size n:
Sample estimate of sx2 is sx2

20. Variance Two basic differences between population and sample variance.
used instead of m
n-1 is used as the denominator rather than n to avoid a biased estimate for sx2
Variance of grouped data

21. Rules for finding the Variance

22. Units of Variance Units of the variance are the same as units on X2.
Units on its positive square root, the standard deviation, sx, are the same as the units of the random variable, X.
A dimensionless measure of dispersion is the coefficient of variation, Cv.

23. Measures of Symmetry Many distributions are not symmetrical
Tailing off to the right or the left is skewing the distribution.
Tailing to the right-positively skewed
Tailing to the left-negatively skewed

24. Skewness
3rd moment about the mean

25. Practical measurements of skewness
One measure of absolute skew is to measure the difference between the mean and the mode.
Not meaningful for comparison sake because it is dependent on units of measure.

26. Pearson’s first coefficient of skewness
Relative measure of skewness more useful for comparison.
Population skewness
Sample skewness

27. Measures of Peakedness (Flatness)
Kurtosis refers to the extent of peakedness of a probability distribution in comparison to the normal distribution.
Kurtosis is the 4th moment about the mean.
Calculate the coefficient of kurtosis, k

28. Kurtosis

29. Covariance
Measure of the linear relationship between two jointly distributed random variables, X and Y.
Covariance is the 1,1 central moment

30. Covariance
If X and Y are independent:
Sample statistic:

31. Correlation Coefficient
Normalized covariance
If X and Y are independent

32. Correlation Coefficient
Measure of how two variables vary together.
A value of r equal to positive one implies that X and Y are perfectly related by Y=a+bX.
Positive values indicate large (small) values of X tend to be paired with large (small) values of Y.
Negative values indicate large (small) values of X tend to be paired with small (large) values of Y.
Two values are uncorrelated ONLY IF r(x,y)=0.
Correlation does NOT equal cause and effect.

33. Correlation coefficient: Linear dependence and functional dependence

34. Other Properties of Moments