Some notes on Probability

Universality of Uniform

Let $F$ be a CDF which is a continuous function and strictly increasing on the support of the distribution. This ensures that the inverse function $F^{-1}$ exists, as a function from $(0,1)$ to $\mathbb{R}$. We then have the following results.

• Let $U\sim \mathrm{Unif}(0,1)$ and $X=F^{-1}(U)$. Then $X$ is an r.v. with CDF $F$.
• Let $X$ be an r.v. with CDF $F$. Then $F(X)\sim \mathrm{Unif}(0,1)$.

• This means, to sample any RV, uniformly pick a value from U(0,1) and then for that value of y, find the corresponding x in CDF of the RV.
• If $X\sim \mathcal{N}\left(\mu ,{\sigma }^2\right)$, then: $$P(|X-\mu |<\sigma )\approx 0.68$$ $$P(|X-\mu |<2\sigma )\approx 0.95$$ $$P(|X-\mu |<3\sigma )\approx 0.997$$
• For any r.v.s $X$,$Y$ and any constant $c$, $$E(X+Y)=E(X)+E(Y)$$ $$E(cX)=cE(X)$$
• Law of the unconscious statistician (LOTUS): If $X$ is a discrete $r$.v. and $g$ is a function from $\mathbb{R}$ to $\mathbb{R}$, then $$E(g(X))=\sum _{x}g(x)P(X=x)$$ where the sum is taken over all possible values of $X$.
• The variance of an r.v. $X$ is $$\mathrm{Var}(X)=E(X-EX{)}^2$$ The square root of the variance is called the standard deviation (SD): $$\mathrm{SD}(X)=\sqrt{\mathrm{Var}(X)}$$ Recall that when we write $E(X-EX{)}^2$, we mean the expectation of the random variable $(X-EX{)}^2$, not $(E(X-EX){)}^2$ (which is 0 by linearity).
• For any r.v. $X$, $$\mathrm{Var}(X)=E\left(X^2\right)-(EX{)}^2$$
• For $X\sim \mathcal{N}\left(\mu ,{\sigma }^2\right)$, we can write $X=\mu +\sigma Z$ with $Z\sim \mathcal{N}(0,1)$, and then $$E(X)=\mu +\sigma \cdot 0=\mu$$ $$\mathrm{Var}(X)={\sigma }^2\mathrm{Var}(Z)={\sigma }^2$$

Law of Large numbers

Let $X_1,X_2,X_3,\dots$ be i.i.d. with mean $\mu$ and variance ${\sigma }^2$. The law of large numbers says that as $n\to \infty$, the sample mean ${\bar{X}}_n$ converges to the constant $\mu$ (with probability 1 ).

• But what is its distribution along the way to becoming a constant? This is addressed by the central limit theorem (CLT), which, as its name suggests, is a limit theorem of central importance in statistics.

Central Limit Theorem

For large $n$, the distribution of ${\bar{X}}_n$ is approximately $\mathcal{N}\left(\mu ,{\sigma }^2/n\right)$. Of course, we already knew from properties of expectation and variance that ${\bar{X}}_n$ has mean $\mu$ and variance ${\sigma }^2/n$; the central limit theorem gives us the additional information that ${\bar{X}}_n$ is approximately Normal with said mean and variance.

• The CLT states that for large $n$, the distribution of ${\bar{X}}_n$ after standardization approaches a standard Normal distribution. By standardization, we mean that we subtract $\mu$, the expected value of ${\bar{X}}_n$, and divide by $\sigma /\sqrt{n}$, the standard deviation of ${\bar{X}}_n$.

Markov Chain

A sequence of random variables $X_0,X_1,X_2,\dots$ taking values in the state space $\{1,2,\dots ,M\}$ is called a Markov chain if for all $n\ge 0$,

$$P\left(X_{n+1}=j\mid X_n=i,X_{n-1}=i_{n-1},\dots ,X_0=i_0\right)=P\left(X_{n+1}=j\mid X_n=i\right)$$

The quantity $P\left(X_{n+1}=j\mid X_n=i\right)$ is called the transition probability from state $i$ to state $j$. (subscripts indicate timestep, values of state space indicate states)

• When referring to a Markov chain we will implicitly assume that it is time-homogeneous, which means that the transition probability $P\left(X_{n+1}=j\mid X_n=i\right)$ is the same for all times $n$. But care is needed, since the literature is not consistent about whether to say “time-homogeneous Markov chain” or just “Markov chain”.
• The above condition is called the Markov property, and it says that given the entire past history , only the most recent term, , matters for predicting.
• The Markov property greatly simplifies computations of conditional probability: instead of having to condition on the entire past, we only need to condition on the most recent value.
• For more on Markov chains, see (Joseph Chang, 2007).

Joseph Chang. (2007). Markov Chains - from Stochastic Processes. http://www.stat.yale.edu/~pollard/Courses/251.spring2013/Handouts/Chang-MarkovChains.pdf