Articles of probability

correlation between a discrete uniform and a continuous uniform rvs

Suppose X and U are independent random variables with P(X = k) =$\frac 1 {N+1}$, k= 0, 1, 2, . . . ,N, and U having a uniform distribution on [0, 1]. Let Y = X + U. a) For y ∈ R, find P(Y ≤ y). b) Find the correlation coefficient between X and […]

Is the set of probability distributions with two mass points and finite 4-th moment compact and closed in the weak{*} topology?

Is the set of probability distributions with two mass points and finite 4-th moment compact and closed in the weak{*} topology. In particular, I would like to see the difference between the following two sets. That is we are looking at the set \begin{align} \mathcal{P}_1=\left \{ F: F=(1-t) \delta_{x_1}+t \delta_{x_2}, \, t\in[0,1], \, x_1,x_2 \in […]

Proof of Logarithmic Mean and Variance

I’m having some issues proving that the mean of a logarithmic distribution is (-1/(ln(q))) * (p/q) and the variance is -p(((p+ln(q))/ ((q)2 ln2 (q))). I have taken the logarithmic distribution, which comes out to be: -log(1-p) = p + p2/2 + p3/3 + … = ∑k=1∞ (-1/(ln(1-p)) * pk/k However, after this step, I’m having […]

Determine the probability that each of the 8 members serves on at least one of the three committees.

A certain group has 8 members. In January, 3 members are selected at random to serve on a committee. In February, 4 members are selected at random and independently of the first selection to serve on another committee. In March, 5 members are selected at random and independently of the previous 2 selections to serve […]

Can all laws be derived from axioms?

I don’t really understand the purpose of an axiom if some laws cannot be derived from them. For example, how is one supposed to prove De Morgan’s laws with only the axioms of probability?

Suggestion on good probability theory book

I am interested in probability theory but I do not have solid background Any suggestions for me to learn probability theory by myself? UPDATE: I have an engineering backgorund. I know the practical stuff but I would like to learn things in a theoretical way.

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it’s the same or am I missing something else. A book I’m reading used a rule called Fatous lemma to give the […]

The minimum of $n$ independent copies of a certain random variable

Consider a random variable $Z$ defined as the sum of two independent copies of the uniform random variable on $[0,1]$. Let $Y$ be the minimum of $n$ independent copies of $Z$. What is the expectation and standard deviation of $Y$? In case an exact answer will be hard to come by, I would be really […]

Limiting probability of Markov chain

So the problem is: Let $X$ be a Markov chain with state space $E = ${a,b} and transition matrix $$p=\begin{bmatrix}0.4 &0.6 \\1 & 0 \end{bmatrix}$$ and suppose that a reward of $g(i,j)$ units is received for every jump from $i$ to $j$ where $$g=\begin{bmatrix}3 &2 \\-1 & 1 \end{bmatrix}$$ Find $$\lim_{n\to \infty}\frac 1{n+1}\sum_{m=0}^{n}g(X_m, X_{m+1})$$ The […]

Almost sure convergence and lim sup

I’m struggling to understand the beginning of the solution to the following exercise: Let $(X_n)_{n\geq 1}$ and $X$ be random variables. Prove that $X_n \to X$ almost surely if and only if for every $\epsilon>0$ $$P(\limsup\limits_{n\to \infty}\{|X_n-X|\geq \epsilon\})=0$$ Solution: Notice that $\omega \in\limsup\limits_{n\to \infty}\{|X_n-X|\geq \epsilon\}$ iff there exist a subsequence $(n_k)_{k\geq 1}$ such that $|X_{n_k}-X|\geq […]