Expected Value & Standard Deviation of Random Variables
Stacey Hancock
1 Definition of Random Variable
When we assign a number to the outcome of a random process, we call this a random variable. For
example, flipping a coin once is a random process, and we could define the random variable Xto take
on the value 0 if the coin lands on tails and 1 if the coin lands on heads. Examples of other random
variables include:
1. The amount of claims (in hundreds of dollars) an insurance company has to pay out to its clients
in a randomly selected month.
2. The time (in hours) a randomly chosen MSU student spends studying for final exams.
3. The number of girls in the next three births at the Bozeman hospital.
4. The number of heads in nine tosses of a coin.
Not only are questions about the distribution of a random variable, or its probabilities, of interest, but
we may want to determine the “average” or expected value of a random variable as well as how far it
tends to vary from its expected value, or its standard deviation. We will only study expected value
and standard deviation for discrete random variables which are random variables whose set of possible
values form a countable list of distinct values. For example, the number of girls in the next three births
at the Bozeman hospital is a discrete random variable since it can only take on the values 0, 1, 2, or
3. Discrete random variables can take on an infinite number of possible values, as long as we can list
them in an ordered list. For example, the number of tosses of a coin until the first head appears is a
discrete random variable with possible values 1, 2, 3, 4, .. . . Random variables that can take on any
value in an interval (e.g., time, length, interest rates, height) are called continuous random variables.
We will use the following notation to specify a probability for a possible outcome of a discrete
random variable:
•X= the random variable (e.g., number of girls in the next three births)
•k= a possible value of the random variable (e.g., 2)
•P(X=k) = the probability that the random variable Xequals the value k
2 Expected Value: What is the average value over many observations
of the random variable?
If we know the probabilities of each of the possible values of a discrete random variable, then we can
calculate the long-run average of the variable, or its mean value over an infinite number of observations
of the random variable. We call this its expected value. Suppose the random variable Xcan take
on possible values k1,k2,k3,... Then we define the expected value of Xas
Mathematical definition:
E(X) = k1×P(X=k1) + k2×P(X=k2) + k3×P(X=k3) + ...
= sum of “value×probability” summed over all possible values
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