Random variable

A random variable $X$ maps each possible outcome $s\in S$ where $S$ is the sample space to a real number. Random variables can represent anything.

Example

If we flip a coin 5 times, we can have a random variable $H$ that is the number of heads. $H$ is a set such that $H\in \{0,1,2,3,4,5\}$, because you can at most flip 5 heads.

We can ask about the probability of an event occurring for a random variable. For example, asking $P(H\ge 4)$ is asking the probability of the event that the number of heads we flip is 4 or greater.

In reality, this is equivalent to $P(\{s\in S\mid H(s)\ge 4\})$. We are considering the set of outcomes such that when it’s mapped to $H$, its value is 4 or greater.

You can also define a random variable by explicitly stating how it maps outcomes to numbers. For example, we can have a $X$ such that $X=1$ if the flip is heads, and $X=0$ if the flip is tails. This would be an Indicator variable.

Discrete random variable

A r.v. is called discrete if there is a countable set of values $V$ that $X$ can equal. $V$ can be a finite set, or a countably infinite set, but not an uncountable set.

The countable set of values $x$ such that the PMF $P(X=x)>0$ is called the support of $X$. This should be $V$.

In problem sets, we need to specify the support whenever we give a PMF for a random variable.

Continuous random variable

Continuous r.v.s can take on any real value in $\mathbb{R}$ for some interval.

It doesn’t make sense to define a PMF for a continuous variable $X$. Asking for a specific value of $X$ makes no sense when it is defined on a continuous interval.

• If $X$ equals the high temperature in Philly, then $P(X=68)=0$. Intuitively, we’re asking for the probability that $X$ equals a single number out of an infinite number of values.
• In fact, for any constant $k$, if $X$ is continuous, then $P(X=k)=0$.

Formal definition

A r.v. $X$ has a continuous distribution if its CDF $F(u)=P(X\le u)$ is:

• Differentiable (and therefore continuous everywhere)
• Or continuous everywhere, and differentiable at all but a finite number of points.

If $X$ has such a continuous distribution, then we say $X$ is a continuous random variable.

Using the continuous distribution instead

We define the support of the random variable as the set of all $u$ such that the PDF of $X$ is greater than zero. That is, $f_X(u)>0$.

The probability that $X$ exists within an interval $[a,b]$, however, is defined. This is just integrating over the PDF from $a$ to $b$.

$$P(a\le X\le b)=F_X(b)-F_X(a)={\int }_a^b{f}_X(t)dt$$

Note that $[a,b]=(a,b)=[a,b)=[a,b)$ because again, the probability that exactly $P(X=a)$ or $P(X=b)$ is zero.

Expectation

Since $X$ is continuous, we now use an integral instead of a summation to find the expected value.

$$\mathbb{E}(X)={\int }_{-\infty }^{\infty }x\cdot f_u(x)dx$$

where $f_u(x)$ is the PDF when $X=x$ (remember that this really is an infinitesimal range).

• Only holds when integral converges absolutely (${\int }_{-\infty }^{\infty }|x|\cdot f_u(x)dx$ is finite). Otherwise, $\mathbb{E}(X)$ is undefined.
• In this course we usually assume $\mathbb{E}(X)$ is well-defined.