A random variable takes numerical values that describe the outcomes of some chance process. The probability distribution of a random variable gives its possible values & their probabilities.
Two types of random variables: Discrete and Continuous
Discrete Random Variables
A discrete random variable X takes a fixed set of possible values with gaps between. The probability distribution of a discrete random variable X lists the values x1 and the probabilities p1.
The probabilities p1 must satisfy two requirements:
- Every probability p1 is a number between 0 and 1
- The sum of the probabilities is 1
To find the probability of any event, add the probabilities p1 of the particular values x1 that make up the event.
Mean (Expected Value) of a discrete random variable
The return of SOCS! Now, we can use our acronym to describe the probability distributions of random variables. We have described some shapes of these distributions already, and outliers are much less common with probability distributions, so let’s look at how to find the center and spread.
We know that the mean x̄ of a set of observations is their average. The mean μx̅ of a discrete random variable X is also an average of the possible values of X but with an important change to take into account: That not all outcomes may be equally likely.
If someone played several games of roulette, we’d call the mean amount the person gained x̄. The means we just calculated are different––those are long-run average gains we’d expect if someone played roulette a very large number of times.
For this reason, the mean of a random variable is often referred to as its expected value.
The mean of a discrete random variable describes the long-run average outcome.
There are two ways of writing the mean of a random variable X. We can use the notation μx̅ as in “mu sub x-bar,” or we can write E(x) as in the “expected value of X.”
To find the mean (expected value) of X, multiply each possible value by its probability, then add all the products.
μx̅ = E(x) = x1p1+x2p2…+xnpn
Notice that this mean value is not a possible value of the random variable X. It’s also not an integer. Remember to think of the mean as a long-run average over many repetitions.
Standard deviation (and variance) of a discrete random variable
Since we’re using the mean as the measure of center for discrete random variables, it then makes sense that we will use standard deviation as well. The definition of the variance for a set of quantitative data. That is, the variance is an “average” of the squared deviation (xi - µx)2 of the values of the variable X from its mean Ux. Each outcome is weighted by its probability to take account of outcomes that are not equally likely.
Variance = Var(X) = Ox2 = (x1-Ux)2p1+ (x2-Ux)2p2+ (x3-Ux)2p3…
Standard deviation of X is Ox = √Var(X)
Remember that discrete random variables often arise from counting things––languages you speak, pets you have, etc. Continuous random variables often come from measuring something––your height, time it takes for you to run a mile, or distance from your house to school.