Contents
Lecture 8: Random variables
 Reading: Cameron [[../handouts/cameron_prob_notes.pdf#page=473.1–3.2, 3.4]], MCS [[../handouts/mcs.pdf#page=82319.1]]
 [[../../2016fa/lectures/2800probability.pdfLast semester's notes]]
 definitions: random variable, PMF, joint PMF, sum/product/etc of RVs, indicator variable, expectation
Random variables
Definition: A (realvalued) random variable
is just a function .Example: Suppose I roll a fair 6sided die. On an even roll, I win $10. On an odd roll, I lose however much money is shown. We can model the experiment (rolling a die) using the sample space
and an equiprobable measure. The result of the experiment is given by the random variable given by , , , , , and .Definition: Given a random variable
and a real number , the poorlynamed event is defined by .This definition is useful because it allows to ask "what is the probability that
?"Definition: The probability mass function (PMF) of
is the function given by .Combining random variables
Given random variables
and on a sample space , we can combine apply any of the normal operations of real numbers on and by performing them pointwise on the outputs of and . For example, we can define by . Similarly, we can define by .We can also consider a real number
as a random variable by defining by . We will use the same variable for both the constant random variable and for the number itself; it should be clear from context which we are referring to.Indicator variables
We often want to count how many times something happens in an experiment.
Example: Suppose I flip a coin 100 times. The sample space would consist of sequences of 100 flips, and I might define the variable
to be the number of heads. For example, , while .A useful tool for counting is an indicator variable:
Definition: The indicator variable for an event
is a variable having value 1 if the happens, and 0 otherwise.The number of times something happens can be written as a sum of indicator variables.
In the coin example, we could define an indicator variable
which is 1 if the first coin is a head, and 0 otherwise (e.g. ). We could define a variable that only looks at the second toss, and so on. Then as defined above can be written as . This is useful because (as we'll see when we talk about expectation) it is often easier to reason about a sum of simple variables (like ) than it is to reason about a complex variable like .Joint PMF of two random variables
We can summarize the probability distribution of two random variables
and using a "joint PMF". The joint PMF of and is a function from and gives for any and , the probability that and . It is often useful to draw a table:

y 


1 
10 

x 
1 
1/3 
1/6 
10 
1/6 
1/3 
Note that the sum of the entries in the table must be one (Exercise: prove this). You can also check that summing the rows gives the PMF of
, while summing the columns gives the PMF of .Expectation
The "expected value" is an estimate of the "likely outcome" of a random variable. It is the weighted average of all of the possible values of the RV, weighted by the probability of seeing those outcomes. Formally:
Definition: The expected value of
, written is given byClaim: (alternate definition of
)Proof sketch: this is just grouping together the terms in the original definition for the outcomes with the same
value.Note: You may be concerned about "
. In discrete examples, almost everywhere, so this sum reduces to a finite or at least countable sum. In nondiscrete example, this summation can be replaced by an integral. Measure theory is a branch of mathematics that puts this distinction on firmer theoretical footing by replacing both the summation and the integral with the socalled "Lebesgue integral". In this course, we will simply use " " with the understanding that it becomes an integral when the random variable is continuous.