1. Recall: The Mean (Expected Value) of a Random Variable

Last lecture, random variables (or if it helps, results of random processes) have means, guess what, they have variances (and therefore standard deviations) too.

2. Random Variables have variances too (4.4)

A random variable has a measure of spread: the variance and its square root, the standard deviation. The variance (and standard deviation) of a random variable measures how far the various outcomes are from the mean or expected value. Just like the definition of variance or standard deviation in Chapter 1.2, there is a similar interpretation -- it's the average squared deviation from the mean.

Symbol: variance is s² and standard deviation is just s. Generally they have a subscript like x to tell you that it is the variance of random variable x or the standard deviation of random variable x.

Notes: some textbooks call the standard deviation of a random variable the "standard error" in order to distinguish this from the standard deviation you learned about in Chapter 1.2 Like the mean in Chapter 1.2 versus the mean of a random variable in Chapter 4.4, this standard deviation in Chapter 4.4 is for a RANDOM PROCESS or EXPERIMENT or a THEORETICAL VALUE. The standard deviation in Chapter 1.2 is for a list of numbers and can be thought of as the sample standard deviation.

3. The variance and standard deviation of a discrete random variable

Just like the mean of a discrete random variable weights each outcome by their respective probabilities, the squared deviations are weighted by their probabilities. For a discrete random variable X, the variance is:

S (x_i - m_x)²p_i

and of course the standard deviation is the square root of the variance.

Example: How we think the Market behaves over a random 3 day period in terms of closing "up" or "not up"

f(x)	0	1	2	3
p(x)	.064	.288	.432	.216

The expected value or mean was 1.8 up days

The variance then is:

(0 - 1.8)²*.064 + (1-1.8)²*.288 + (2-1.8)²*.432 + (3 - 1.8)²*.216 = .72 "up "days squared

The standard deviation is a little easier to interpret

Square root of .72 = .8485 so .8485 "up" days

Interpreting these two will be a little easier in coming days. For now, interpret them like the mean and standard deviation you are familiar with, just keep in mind that these are the means and standard deviations of random variables.

4. The variance of a continuous random variable

Like the mean of a continuous random variable, it will be given to you. You won't be asked to calculate the variance of a continuous random variable. Again, normally distributed random variables are the best examples of a continuous random variable. You are given a mean and a standard deviation and will be expected to make statements about the distribution.

5. Rules and properties of the variance of a random variable

a. If you multiply a random variable by a constant b, the variance will be multiplied by b-squared (b²). If you add a constant to a random variable, the variance will not be affected by addition.

b. If you have two independent random variables X and Y then regardless of whether you are interested in the sum of the two (X+Y) or the difference between the two (X-Y), the variances MUST be ADDED together to get the combined variance. Do not subtract variances.

Note: You cannot add standard deviations to get a combined standard deviation, you must find the variances, add those together and then take the square root to find the combined standard deviation.

From last time: You sell homes

Homes Sold	0	1	2	3	4
Probability	.1	.5	.3	.1	0

From last time: You partner sells homes too

Homes Sold	0	1	2	3	4
Probability	.1	.1	.5	.2	.1

Your mean weekly homes sold is 1.4, your partner's is 2.1

Your combined mean is 1.4+2.1 = 3.5

Your variance is: (0-1.4)² *.1 + (1-1.4)² *.5 + (2-1.4)² *.3 + (3-1.4)² *.1 + (4-1.4)² *0 = .64

Your standard deviation is = .80

Your partner's variance is: (0-2.1)² *.1 + (1-2.1)² *.1 + (2-2.1)² *.5 + (3-2.1)² *.2 + (4-2.1)² *.1 = 1.09

Your partner's standard deviation is 1.044

Your combined standard deviation is 1.315

Your combined mean earnings (from last time) were 19,900 because you got 10,000 per home sold less 2500 and your partner got 4,000 per home sold. If we use that information the combined variance is:

10000²*.64 + 4000²*1.09 = 81,440,000

and the standard deviation will be $9,024.41. So you could think that in any randomly chosen week, you probably sell 3.5 homes on average, earn $19,900 on average with a standard deviation of about 9,024 dollars.