Random Variables and their Distributions
The condition in Definition 5 is necessary to compute
. This requirement also let us compute
for most sets by leveraging the fact that
is closed under complements, unions, and intersections. For example, we can also compute
. In this sense, the property binds the probability space to the random variable.
Definition 5 also implies that random variables satisfy particular algebraic properties. For example, if
are random variables, then so are
Although random variables are defined based on a probability space, it is often most natural to model problems without explicitly specifying the probability space. This works so long as we specify the random variables and their distribution in a “consistent” way. This is formalized by the so-called Kolmogorov Extension Theorem but can largely be ignored.
Roughly speaking, the distribution of a random variable gives an idea of the likelihood that a random variable takes a particular value or set of values.
Continuous random variables are largely similar to discrete random variables. One key difference is that instead of being described by a probability “mass”, they are instead described by a probability “density”.
Observe that if a random variable
is continuous, then the probability that it takes on a particular value is zero.
Note that by the Kolomogorov axioms,
must satisfy three properties:
- 1.is non-decreasing.
- 3.is right continuous.
It turns out that if we have any function
that satisfies these three properties, then it is the CDF of some random variable on some probability space. Note that
gives us an alternative way to define continuous random variables. If
is absolutely continuous, then it can be expressed as
for some non-negative function
, and this is the PDF of a continuous random variable.
Often, when modeling problems, there are multiple random variables that we want to keep track of.
Note that it is possible for
to be continuous and
to be discrete (or vice versa).
Just like independence, we can extend the notion of conditional probability to random variables.
Often, we need to combine or transform several random variables. A derived distribution is the obtained by arithmetic of several random variables or applying a function to several (or many) random variables. Since the CDF of a distribution essentially defines that random variable, it can often be easiest to work backwards from the CDF to the PDF or PMF. In the special case where we want to find
for a function
Another special case of a derived distribution is when adding random variables together.
Expectation has several useful properties. If we want to compute the expectation of a function of a random variable, then we can use the law of the unconscious statisitician.
Another useful property is its linearity.
Sometimes it can be difficult to compute expectations directly. For disrete distributions, we can use the tail-sum formula.
When two random variables are independent, expectation has some additional properties.
Earlier, we saw that we find a derived distribution by transforming and combining random variables. Sometimes, we don’t need to actually compute the distribution, but only some of its properties.
It turns out that we can encode the moments of a distribution into the coefficients of a special power series.
Notice that if we apply the power series expansion of
, we see that
Thus the nth moment is encoded in the coefficients of the power series and we can retrieve them by taking a derivative:
Another interesting point to notice is that for a continuous random variable
is the Laplace transform of the distribution over the real line, and for a discrete random variable,
is the Z-transform of the distribution evaluated along the curve at
This provides another way to compute the distribution for a sum of random variables because we can just multiply their MGF.
Bernoulli random variables are good for modeling things like a coin flip where there is a probability of success. Bernoulli random variables are frequently used as indicator random variables
When paired with the linearity of expectation, this can be a powerful method of computing the expectation of something.
A binomial random variable can be thought of as the number of successes in
trials. In other words,
By construction, if
are independent, then
Geometric random variables are useful for modeling the number of trials required before the first success. In other words,
A useful property of geometric random variables is that they are memoryless:
Poisson random variables are good for modeling the number of arrivals in a given interval. Suppose you take a given time interval and divide it into
chunks where the probability of arrival in chunk
. Then the total number of arrivals
is distributed as a Binomial random variable with expectation
. As we increase
to infinity but keep
fixed, we arrive at the poisson distribution.
A useful fact about Poisson random variables is that if
are independent, then
The CDF of a uniform distribution is given by
Exponential random variables are the only continuous random variable to have the memoryless property:
The CDF of the exponential distribution is given by
The standard normal is
, and it has the CDF
There is no closed from for
. It turns out that every normal random variable can be transformed into the standard normal (i.e
). Some facts about Gaussian random variables are
- 1.Ifare independent, then.
- 2.Ifare independent andare independent, then bothandare Gaussian with the same variance.
Jointly Gaussian Random Varables, also known as Gaussian Vectors, can be defined in a variety of ways.
In addition to their many definitions, jointly gaussian random variables also have interesting properties.
Theorem 11 tells us that each entry in Gaussian Vector can be thought of as a “noisy” version of the others.
One way to understand random variables is through linear algebra by thinking of them as vectors in a vector space.
Inner products spaces are equipped with the norm
Loosely, completeness means that we can take limits of without exiting the space. It turns out that random variables satisfy the definition of a Hilbert Space.
Hilbert spaces are important because they provide a notion of geometry that is compatible with our intuition as well as the geometry of
(which is a Hilbert Space). One geometric idea is that of orthogonality. Two vectors are orthogonal if
. Two random variables will be orthogonal if they are zero-mean and uncorrelated. Using orthogonality, we can also define projections.
Theorem 13 is what gives rise to important properties like the Pythogorean Theorem for any Hilbert Space.
Suppose we had to random variables
. What happens if we try and project one onto the other?
Thus, the conditional expectation is the function of
that is closest to
. It’s interpretation is that the expectation of
can change after observing some other random variable
. To find
, we can use the conditional distribution of
is a function of the random variable
, meaning we can apply Theorem 6.
Alternatively, we could apply lineary of expectation to Definition 34 to arrive at the same result. If we apply Theorem 15 to the function
, then we can see that
Just as expectation can change when we know additional information, so can variance.
Conditional variance is a random variable just as expectation is.
The second term in the law of total variance (
) can be interpreted as on average, how much uncertainty there is in
given we know