Tagged Questions

Use this tag only if your question is about the modern theoretical footing for probability, for example probability spaces, random variables, law of large numbers, and central limit theorems. Use [tag:probability] instead for specific problems and explicit computations. Use [tag:probability-...

2k views

What does the -log[P(X)] mean in the calculation of entropy?

The entropy (self information) of a discrete random variable X is calculated as: $$H(x)=E(-log[P(X)])$$ What does the -log[P(X)] mean? It seems to be something like ""the self information of each ...
897 views

668 views

Limit of sums of iid random variables which are not square-integrable

The Central Limit Theorem tells us that for an iid sequence of random variables $(X_n)_{n\geq 0}$ of finite variance $\sigma^2$ and zero mean $$\lim_{n\to\infty}\frac{S_n}{\sqrt{n}}=^d N(0,\sigma^2)$$...
1k views

A Good Book for Mathematical Probability Theory [duplicate]

I am from mathematical background, and I always hated the way they teach elementary probability theory in schools without giving any clue about measure theory. I want a theoretical book in ...
803 views

When the sum of independent Markov chains is a Markov chain?

I try to find as much as possible cases, when the chain $Z(t) = |X_1(t)-X_2(t)|$ is Markov, where $X_1(t)$ and $X_2(t)$ are independent, discrete-time and space, preferably non-homogeneous Markov ...
1k views

Combinations of characteristic functions: $\alpha\phi_1+(1-\alpha)\phi_2$

Suppose we are given two characteristic functions: $\phi_1,\phi_2$ and I want to take a weighted average of them as below: $\alpha\phi_1+(1-\alpha)\phi_2$ for any $\alpha\in [0,1]$ Can it be proven ...
608 views

607 views

Different of mapsto and right arrow

Could someone please explain to me what is the difference in the two arrows$$\rightarrow$$ and $$\mapsto$$ For example in Probability wih Martingales (Willams) Thank you.
154 views

432 views

Can you make money on coin tosses when the odds are against you?

The strategy Given an initial investment $n$ dollars and a "bet buffer" $b$. Calculate the bet size $x=\left\lfloor\frac{n}{2^b-1}\right\rfloor$ dollars. Wager $x$ dollars on random variable $C$ ...
929 views