Mathematics Prelims

November 1, 2008

Distribution of a Random Variable

Filed under: Measure Theory,Probability Theory — cjohnson @ 12:27 pm

Let (\Omega, \mathcal{F}, P) be a probability space and X : \Omega \to \mathbb{R} a random variable.  Define a function F(x) as the probability X takes on a value less than or equal to x.

\displaystyle F(x) := P(X^{-1}((-\infty, x])

That is, F(x) is the measure of the set of \omega \in \Omega such that X(\omega) \leq x.  Notice that F(x) is an increasing function as a \leq b implies (-\infty, a] \subseteq (-\infty, b] and so

\displaystyle X^{-1}((-\infty, a]) = \{ \omega \in \Omega : X(\omega) \leq a \}

\displaystyle \, \, \subseteq \{ \omega \in \Omega : X(\omega) \leq b \}

\displaystyle \, \, = X^{-1}((-\infty, b])

And since a measure is monotonic (i.e., A \subseteq B \implies P(A) \leq P(B)), we have that F(a) \leq F(b).

Recall that if (E_n)_{n \in \mathbb{N}} is a decreasing sequence of measurable sets (i.e., E_n \supseteq E_{n+1}), then P\left(\bigcap_{n \in \mathbb{N}} E_n\right) = \lim_{n \to \infty} P(E_n), provided there exsists an E_n with P(E_n) finite (in our case P is a probability measure, so this is given to us anyway).  We can use this fact to show that our F function is right-continuous.

Let (x_n)_{n \in \mathbb{N}} be a real-valued sequence that decreases to x.  Then we have that

\displaystyle \lim_{n \to \infty} F(x_n) = \lim_{n \to \infty} P(X^{-1}((-\infty, x_n]))

\displaystyle = P \left( \bigcap_{n = 1}^\infty X^{-1}((-\infty, x_n])\right)

\displaystyle = P \left( X^{-1} \left( \bigcap_{n=1}^\infty (-\infty, x_n] \right) \right)

\displaystyle = P(X^{-1}((-\infty, x]))

\displaystyle = F(x)

This shows that \displaystyle \lim_{x \to x_0^+} F(x) = F(x_0) and so F is right-continuous.

Additionally, \lim_{x \to \infty} F(x) = 1 and \lim_{x \to -\infty} F(x) = 0.

The function F that we’ve described is known as the cumulative distribution function (or just distribution function) of our random variable X.  If F(x) only takes on countably many values, then we say that X is a discrete random variable.  If F(x) is a continuous function, we say X is a continuous random variable.

Now that we’ve seen how to construct this distribution function given a random variable X, the natural question to ask is if we have a function that satisfies the properties of a distribution that we’ve listed, is there a corresponding random variable?

Suppose that F(x) is an increasing, right-continuous function with \lim_{x \to -\infty} F(x) = 0 and \lim_{x \to \infty} F(x) = 1.  Using the ideas from the Lebesgue-Stieltjes measure article, we have that F(x) gives us a measure and sigma-algebra on \mathbb{R}.  Let P_F and \mathcal{F}_F be the measure and sigma-algebra, respectively, that we get from F(x).  It follows from the fact that \lim_{x \to \infty} F(x) = 1 and \lim_{x \to -\infty} F(x) = 0 that P_F(\mathbb{R}) = 1, and so (\mathbb{R}, \mathcal{F}_F, P_F) is a probability space.

Define X : \mathbb{R} \to \mathbb{R} as x \mapsto x.  Note that this is a random variable as

\displaystyle X^{-1}((-\infty, x]) = (-\infty, x]

To see that X is a measurable function (random variable), let A \subseteq \mathbb{R} and \epsilon > 0.  There exists a sequence (I_n)_{n \in \mathbb{N}} of intervals that covers A and

\displaystyle P_F^*(A) \leq \sum_{n=1}^\infty \ell_F(I_N) \leq P_F^*(A) + \epsilon

Define J_n := I_n \cap (-\infty, t] and K_n := I_n \cap (t, \infty); note that \bigcup_n J_n and \bigcup_n K_n cover A \cap (-\infty, t) and A \cap (t, \infty), respectively.  Also notice that \ell_F(I_n) = \ell_F(J_n) + \ell_F(K_n).  This gives us

\displaystyle P_F^*(A) \leq \sum_{n=1}^\infty \ell_F(J_n) + \sum_{n=1}^\infty \ell_F(K_n) \leq P_F^*(A) + \epsilon

\displaystyle \implies P_F^*(A) \leq P_F^*(A \cap (-\infty, t]) + P_F^*(A \cap (t, \infty)) \leq P_F^*(A) + \epsilon

\displaystyle \implies P_F^*(A) = P_F^*(A \cap (-\infty, t]) + P_F^*(A \cap (-\infty, t]^\complement)

so X is a measurable function (random variable).  This means that any function F which is increasing, right-continuous, and \lim_{x \to -\infty} F(x) = 0 and \lim_{x \to \infty} F(X) = 1, is the distribution of some random variable.

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