Articles of probability

Number of primes less than 2n

A series of questions. Explanations would be useful. I have done the first four parts. I am confused on how to go about the last two. Show that for any prime $p$ the largest power of $p$ that divides $n!$ is $$ \left\lfloor \frac{n}{p}\right\rfloor +\left\lfloor \frac{n}{p^2} \right\rfloor +\cdots +\left\lfloor\frac{n}{p^r}\right\rfloor$$ where $p^r\le n < p^{r+1}$ Use […]

Distribution of $r$ balls into $n$ cells leaving none of the cells empty.

There are $n$ distinguishable cells and $r$ distinguishable balls. Let $A(r,n)$ be the number of distributions leaving none of the cells empty. Show by combinatorial argument that $A(r, n+1) = \sum_{k=1}^r rC_k \times A(r-k, n)$ Note: This is a question from “An Introduction To Probability: Theory and its Applications” by William Feller. I have tried […]

Conditional Probability: $P(A|B) = P(A|C_i)P(C_i|B)$

basic refresher question about Conditional Probability: Can someone please provide a basic proof of the following identity: In the discrete case: $P(A|B) = \sum_{i} P(A|C_i)P(C_i|B)$ In the continuous case: $P(A|B) = \int P(A|C)P(C|B)dC$

Joint distribution of non homogeneous Poisson event times?

I am trying to calculate the density of $(T_1,T_2)$ where $T_1$ is the time of the first event and $T_2$ is the time of the second event. I have been looking at the Wiki article on Poisson process and while it has been helpful, I haven’t been able to figure out how to apply it […]

The variance of the expected distortion of a linear transformation

Let $A: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation. I am interested in the “average distortion” caused by the action of $A$ on vectors. (i.e stretching or contraction of the norm). Consider the uniform distribution on $\mathbb{S}^{n-1}$, and the random variable $X:\mathbb{S}^{n-1} \to \mathbb{R}$ defined by $X(x)=\|A(x)\|_2^2$. It is easy to see that $E(X)=\frac{1}{n}\sum_{i=1}^n \sigma_i^2$, […]

Solving Addition Rule other way

I know the General Addition Rule for Two Events is $$P (A\cup B) = P (A) + P(B) – P(A\cap B)$$ I know the proof where ${R_1} = A\cap B, R_2 = A\cap B’, R_3 = A’\cap B,$ and $R_4 = A’\cap B’.$ The $R_i$’s are a partition of a sample space S and they […]

Can the sum of two i.i.d. random variables be uniformly distributed?

Let $U$ be a uniform random variable on the interval $[0,1]$. It is exceedingly unlikely that $U$ can be written as a sum $U = X + Y$ where $X$ and $Y$ are independent identically distributed random variables. Consider, for example, the discrete analogue where it is very clear that no such decomposition is possible. […]

Sum of i.i.d. random variables is a markov chain

I think I have some problem understanding markov chains, because we defined them as abstract objects but our professor does proofs with them as if they where just elementary conditional probabilities. This is our definition of a markov chain: Given prob. space $(\Omega, \mathcal{A}, \mathbb{P})$, standard borel space $(S, \mathcal{S})$ and a sequence of random […]

the probability density function (PDF) of concatenation of two Gaussian variables

Gaussian variable $x$ follows from $N(u_x,\sigma_x^2)$ and $y$ follows from $N(u_y,\sigma_y^2)$. Assume we have the vector $\bf{z}=[x,y]^T\in R^2$, then it seems that no matter whether $x$ and $y$ are independent or not, we always have that $\bf{z}$ also follows from the Gaussian distribution $N([u_x,u_y]^T, Cov([x,y]^T))$, where $Cov$ means the covariance. The above claim is reformulated […]

Probability of a four letter word from a sequence of n random letters

I know this is easy, but my high school maths has failed me. Question: I generate an 8 letter random string. What is the probability that within these 8 letters I will find a particular 4 letter word? Each letter is A-Z. Repeats are allowed. What are the chances my string will contain the word […]