I'm introduced to the Radon-Nikodym derivative in the following way:
Probability space $(\Omega, \mathcal{F}, \mathbb{P})$: $\mathbb{P}$ a probability measure on $\Omega$. A random variable $X$ is a deterministic function $X:\Omega\to\mathbb{R}$. Distribution of $X$: determined by how $\mathbb{P}$ assigns probabilities to subsets $\Omega$ and how $X$ maps those to subsets of $\mathbb{R}$.
An example: $\Omega = [0,1]$, and $\mathbb{P}$ is uniform on $\Omega$, i.e., for $0\leq a\leq b\leq 1$: $\mathbb{P}[a,b] = b-a$.
Define $X:[0,1]\to\mathbb{R}$ by $X(w) = -\log w$. Under $\mathbb{P}$ the rv $X$ has an $\operatorname{Exp}(1)$ distribution. Alternative measure $Q$ assigns $Q[a,b] = b^2 – a^2$. Under $Q$ the rv $X$ has an $\operatorname{Exp}(2)$ distribution: $$Q\{X\leq x\} = Q\{w:-\log w\leq x\} = Q[e^{-x}, 1] = 1^2 – (e^{-x})^2 = 1 – e^{-2x}$$The Radon-Nikodym derivative of $Q$ w.r.t. $\mathbb{P}: \,\,\dfrac{dQ}{d\mathbb{P}}(w) = 2w.$
Now there are a couple of things that I don't understand about this example:
- Why is the notation of a set used when writing down the cdf of $Q$? We have that $Q\{X\leq x\} = 1 – e^{-2x}$, why don't we just write $Q(X\leq x) = 1 – e^{-2x}$?
- How should I calculate the Radon-Nikodym derivative? I don't understand how you would arrive at $\dfrac{dQ}{d\mathbb{P}}(w) = 2w$? Both $\mathbb{P}$ and $Q$ take intervals as arguments right? Could someone show me how you would compute this derivative?
Thanks!
Best Answer
There is no difference between the notations $Q(X\leq x)$ and $Q\{X\leq x\}$. For the second question the Radon Nikodym derivative $f$ of $Q$ with respect to $P$ is defined by the equation $Q(E)=\int_E fdP$ for every measurable set $E$. In order find what this $f$ is it is enough consider the sets $E=[o,x]$ where $0\leq x \leq 1$. Thus we have to find $f$ such that $1-e^{-2x} =Q(X\leq x)=\int_{[0,x]} f(t)dt$. [ Note that $P$ is just the uniform measure (i.e. the Lebesgue measure on $[0,1]$ so $\int_E fdP=\int_E f(y)dy)$. To find $f$ from the equation $1-e^{-2x} =\int_0^{x} f(y)dy$ simply differentiate both sides with respect to $x$. Hence $f(x)=2e^{-2x}$.