9 4.2 Measures of General DRVs
Once we know how to work with Discrete Random Variables we may be interested in some other measures such as the mean, variance, and standard deviation. The ideas here are slightly different than we have seen before within our new context of Random Variables.
The Expected Value (Mean) of a Discrete Random Variable
Recall the Law of Large Numbers which states as the number of trials in a probability experiment increases our results become closer to what we expect. When evaluating the long-term results of statistical experiments, we often want to know the “average” outcome. This “long-term average” is known as the mean or expected value of the random variable and is denoted by the Greek letter μ or E[X] in the context of random variables. In other words, after conducting many trials of an experiment, you would expect this average value.
To find the expected value or long term average we simply multiply each value of the random variable by its probability and add the products.
Mean or Expected Value: [latex]\mu=\underset{x\in X}{{\sum }^{\text{}}}xP\left(x\right)[/latex]
Example
A men’s soccer team plays soccer zero, one, or two days a week. The probability that they play zero days is 0.2, the probability that they play one day is 0.5, and the probability that they play two days is 0.3. Find the long-term average or expected value, μ, of the number of days per week the men’s soccer team plays soccer.
To do the problem, first let the random variable X = the number of days the men’s soccer team plays soccer per week. X takes on the values 0, 1, 2. Construct a PDF table adding a column x*P(x). In this column, you will multiply each x value by its probability.
| x | P(x) | x*P(x) |
|---|---|---|
| 0 | 0.2 | (0)(0.2) = 0 |
| 1 | 0.5 | (1)(0.5) = 0.5 |
| 2 | 0.3 | (2)(0.3) = 0.6 |
What is the expected value?
Your turn!
A hospital researcher is interested in the number of times the average post-op patient will ring the nurse during a 12-hour shift. For a random sample of 50 patients, the following information was obtained. What is the expected value?
| x | P(x) |
|---|---|
| 0 | P(x = 0) = [latex]\frac{4}{50}[/latex] |
| 1 | P(x = 1) = [latex]\frac{8}{50}[/latex] |
| 2 | P(x = 2) = [latex]\frac{16}{50}[/latex] |
| 3 | P(x = 3) = [latex]\frac{14}{50}[/latex] |
| 4 | P(x = 4) = [latex]\frac{6}{50}[/latex] |
| 5 | P(x = 5) = [latex]\frac{2}{50}[/latex] |
The Variance and Standard Deviation of a Discrete Random Variable
Like data, probability distributions have standard deviations. To calculate the standard deviation (σ) of a probability distribution, find each deviation from its expected value, square it, multiply it by its probability, add the products, and take the square root.
Finding the variance, σ² or V[X], and standard deviation, σ or SD[X] of a random variable starts similar to what we have seen before but differs at step 4:
- Find the mean
- Subtract the mean from each value of x to get your deviations
- Square each deviation
- Multiply each squared deviation by it’s probability, P(x)
- Sum each of the products
At this point you now have the variance then can of course take the square root of the variance to get your standard deviation. The formula looks like this:
[latex]\sigma = \sqrt{\underset{x\in X}{{\sum }^{\text{}}}{\left(x-\mu \right)}^{2}P\left(x\right)}[/latex]
Example
Find the expected value of the number of times a newborn baby’s crying wakes its mother after midnight. The expected value is the expected number of times per week a newborn baby’s crying wakes its mother after midnight. Calculate the standard deviation of the variable as well.
| x | P(x) | x*P(x) | (x – μ)2 ⋅ P(x) |
|---|---|---|---|
| 0 | P(x = 0) = [latex]\frac{2}{50}[/latex] | (0)[latex]\left(\frac{2}{50}\right)[/latex]= 0 | (0 – 2.1)2 ⋅ 0.04 = 0.1764 |
| 1 | P(x = 1) = [latex]\left(\frac{11}{50}\right)}[/latex] | (1)[latex]\left(\frac{11}{50}\right)[/latex]= [latex]\frac{11}{50}[/latex] | (1 – 2.1)2 ⋅ 0.22 = 0.2662 |
| 2 | P(x = 2) = [latex]\frac{23}{50}[/latex] | (2)[latex]\left(\frac{23}{50}\right)[/latex] = [latex]\frac{46}{50}[/latex] | (2 – 2.1)2 ⋅ 0.46 = 0.0046 |
| 3 | P(x = 3) = [latex]\frac{9}{50}[/latex] | (3)[latex]\left(\frac{9}{50}\right)[/latex] = [latex]\frac{27}{50}[/latex] | (3 – 2.1)2 ⋅ 0.18 = 0.1458 |
| 4 | P(x = 4) = [latex]\frac{4}{50}[/latex] | (4)[latex]\left(\frac{4}{50}\right)[/latex] = [latex]\frac{16}{50}[/latex] | (4 – 2.1)2 ⋅ 0.08 = 0.2888 |
| 5 | P(x = 5) = [latex]\frac{1}{50}[/latex] | (5)[latex]\left(\frac{1}{50}\right)[/latex]= [latex]\frac{5}{50}[/latex] | (5 – 2.1)2 ⋅ 0.02 = 0.1682 |
a. Add the values in the third column of the table to find the expected value of X.
b. Use μ to complete the table. The fourth column of this table will provide the values you need to calculate the standard deviation. For each value x, multiply the square of its deviation by its probability. (Each deviation has the format x – μ).
c. Add the values in the fourth column of the table:
d. The standard deviation of X is the square root of this sum.
e. The mean, μ, of a discrete probability function is the expected value.
f. The standard deviation, Σ, of the PDF is the square root of the variance.
When all outcomes in the probability distribution are equally likely, these formulas coincide with the mean and standard deviation of the set of possible outcomes.
Your turn!
On May 11, 2013 at 9:30 PM, the probability that moderate seismic activity (one moderate earthquake) would occur in the next 48 hours in Japan was about 1.08%. You bet that a moderate earthquake will occur in Japan during this period. If you win the bet, you win $100. If you lose the bet, you pay $10. Let X = the amount of profit from a bet. Find the mean and standard deviation of X.
Note on Calculations
Generally for probability distributions, we use a calculator or a computer to calculate μ and σ to reduce roundoff error. For many special cases of probability distributions, there are short-cut formulas for calculating μ, σ, and associated probabilities. We will see some of these in the future.
Mean of a random variable