2. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
The mean or expected value of a random variable X is important because it describes the center of the probability distribution.
However, the mean does not give adequate description of the shape and variability in the distribution.
Distribution with equal means but different dispersions (variability)
The most important measure of variability of a random variable X is obtained by letting g(X) = (X– μ)2.
This variability measure is referred to as the variance of the random variable X or the variance of the probability distribution of X. It is denoted by Var(X) or the symbol , or simply by .

3. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Let X be a random variable with probability distribution f(x) and mean μ. The variance of X is
if X is discrete, and
if X is continuous.
The positive square root of the variance, σ, is called the standard deviation of X.

4. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Company A
Let the random variable X represent the number of cars that are used for official business purposes on any given workday. The probability distribution for company A and company B are
Company B
Show that the variance of the probability distribution for company B is greater than that of company A.
Clearly, the variance of the number of cars that are used for official business purposes is greater for company B than for company A.

5. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
The variance of a random variable X is also given by
Let the random variable X represent the number of defective parts for a machine when 3 parts are sampled from a production line and tested. The following is the probability distribution of X
Calculate the variance σ2.

6. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
The weekly demand for a drinking-water product, in thousands liters, from a local chain of efficiency stores, is a continuous random variable X having the probability density
Find the mean and variance of X.

7. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Let X be a random variable with probability distribution f(x). The variance of the random variable g(X) is
if X is discrete, and
if X is continuous.

8. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Calculate the variance of g(X) = 2X + 3, where X is a random variable with probability distribution given as

9. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Let X be a random variable with density function
Find the variance of the random variable g(X) = 4X + 3 if it is known that the expected value of g(X) = 8.

10. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Let X and Y be a random variables with probability distribution f(x, y). The covariance of the random variables X and Y is
if X and Y are discrete, and
if X and Y are continuous.
σXY >0,
Positive correlation
σXY <0
Negative correlation

11. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
The covariance of two random variables X and Y with means μX and μY, respectively, is given by

12. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Referring back again to the “ballpoint pens” example, find the covariance of X and Y.
See again Lecture 4

13. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
The fraction X of male runners and the fraction Y of female runners who compete in marathon races is described by the joint density function
Find the covariance of X and Y

14. Variance and Covariance
Chapter 4.2
Variance and Covariance
Variance and Covariance
Although the covariance between two random variables does provide information regarding the nature of the relationship, the magnitude of σXY does not indicate anything regarding the strength of the relationship, since σXY is not scale free.
This means, that its magnitude will depend on the units measured for both X and Y.
There is a scale-free version of the covariance called the correlation coefficient, that is used widely in statistics.
Let X and Y be random variables with covariance σXY and standard deviation σX and σY, respectively. The correlation coefficient X and Y is

15. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
If a and b are constant, then
Applying theorem to the discrete random variable g(X) = 2X – 1, rework the carwash example.

16. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
Let X be a random variable with density function
Find the expected value of g(X) = 4X + 3 by using the theorem presented recently.

17. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
The expected value of the sum or difference of two or more functions of a random variable X is the sum or difference of the expected values of the functions. That is
Let X be a random variable with probability distribution as given next. Find the expected value of Y = (X – 1)2.

18. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
The weekly demand for a certain drink, in thousands of liters, at a chain of convenience stores is a continuous random variable g(X) = X2 + X – 2, where X has the density function
Find the expected value for the weekly demand of the drink.

19. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
The expected value of the sum or difference of two or more functions of a random variables X and Y is the sum or difference of the expected values of the functions. That is
Let X and Y be two independent random variables. Then

20. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
In producing gallium-arsenide microchips, it is known that the ratio between gallium and arsenide is independent of producing a high percentage of workable wafers, which are the main components of microchips.
Let X denote the ratio of gallium to arsenide and Y denote the percentage of workable microwafers retrieved during a 1-hour period. X and Y are independent random variables with the joint density being known as
Illustrate that E(XY) = E(X)E(Y).

21. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
Hence, it is proven that

22. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
If a and b are constant, then
If X and Y are random variables with joint probability distribution f(x, y), then

23. Means of Linear Combinations of X
Chapter 4.3
Means and Variances of Linear Combinations of Random Variables
Means of Linear Combinations of X
If X and Y are random variables with variances , , and covariance σXY = –2, find the variance of the random variable Z = 3X – 4Y + 8.
Let X and Y denote the amount of two different types of impurities in a batch of a certain chemical product. Suppose that X and Y are independent random variables with variances and Find the variance of the random variable Z = 3X – 2Y + 5.

24. Chapter 4.4
Chebyshev’s Theorem
Chebyshev’s Theorem
As we already discussed, the variance of a random variable tells us something about the variability of the observation about the mean.
If a variable has a small variance or standard deviation, we would expect most of the values to be grouped around the mean.
The probability that a random variable assumes a value within a certain interval about the main is greater in this case.
If we think of probability in terms of area, we would expect a continuous distribution with a small standard deviation to have most of its area close to μ.
Variability of continuous observations about the mean

25. Chapter 4.4
Chebyshev’s Theorem
Chebyshev’s Theorem
We can argue the same way for a discrete distribution. The spread out of an area in the probability histogram indicates a more variable distribution of measurements or outcomes.
Variability of discrete observations about the mean

26. Chapter 4.4
Chebyshev’s Theorem
Chebyshev’s Theorem
A Russian mathematician P. L. Chebyshev discovered that the fraction of the area between any two values symmetric about the mean is related to the standard deviation.
|Chebyshev’s Theorem| The probability that any random variable X will assume a value within k standard deviations of the mean is at least 1 – 1/k2.
That is
Chebyshev’s Theorem holds for any distribution of observations and, for this reason, the results are usually weak.
The value given by the theorem is a lower bound only. Exact probabilities can only be determined when the probability distribution is known.
The use of Chebyshev’s Theorem is relegated to situations where the form of the distribution is unknown.

27. Chebyshev’s Theorem and Normal Distribution
Chapter 4.4
Chebyshev’s Theorem
Chebyshev’s Theorem and Normal Distribution

28. Chapter 4.4
Chebyshev’s Theorem
Chebyshev’s Theorem
A random variable X has a mean μ = 8, a variance σ2 = 9, and an unknown probability distribution. Find
P(–4 < X < 20)
P(|X – 8| ≥ 6)

29. Probability and Statistics
Homework 5
For the joint probability distribution of the two random variables X and Y as given in the following figure, calculate the covariance of X and Y. (Mo.E5.27 p.0172)
The photoresist thickness in semiconductor manufacturing has a mean of 10 micrometers and a standard deviation of 1 micrometer. Bound the probability that the thickness is less than 6 or greater than 14 micrometers. (Mo.S5.25 p05.15)