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:
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) = |
| 1 | P(x = 1) = |
| 2 | P(x = 2) = |
| 3 | P(x = 3) = |
| 4 | P(x = 4) = |
| 5 | P(x = 5) = |
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