Articles of probability

a probability question using percentages

This question is confusing me as I am not used to seeing percentages in a possibility question. in a large insurance agency – 60% of the customers have automobile insurance – 40% of the customers have homeowners insurance – 75% of the customers have on type or the other or both a) find the proportion […]

Cover times and hitting times of random walks

Consider a random walk on an undirected graph which, at each step, moves to a uniformly random neighbor. Define $T(u,v)$ to be the expected time until such a walk, starting from $u$, arrives at $v$, and let $T = \max_{u,v} T(u,v)$. Define $G(u)$ to be the expected time until such a walk, starting from $u$, […]

How to test a computer language random number generator

In a computer program I’ve written I am using the computer languages random number function to simulate the tossing of a coin. The random number function returns either -1 (= tails) or +1 (= heads). My computer program does 25 runs of 100,000,000 tosses and sums the (-1’s and +1’s to calculate the excess of […]

Probabilistic Conditioning. Please help me finish the solution to the problem.

I have to solve the following problem: The real random variables $X$ and $Y$ are independent and have a uniform distribution $U([0,1])$. Find $$\mathbb{E}\left( \frac{3 X-Y+1}{\sqrt{X+Y+1}} | \quad e^{X+Y}-20 \log(X+Y) \right)$$ Answer: $\sqrt{X+Y+1}$. My solution: In the condition there is the function $f(y)=e^y-20 \log(y)$. Its derivative is as follows: $f'(y)=e^y-\frac{20}{y}$. Let us note that for […]

Selecting n matches from two pockets.

Setting An eminent mathematician fuels a smoking habit by keeping matches in both trouser pockets. When impelled by need he reaches a hand into a randomly selected pocket and grubs about for a match. Suppose he starts with n matches in each pocket. determine the probability that at the moment the first pocket is emptied […]

Prove or disprove that the given expression is “always” positive

I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that $$\small\int_{\mathbb{R}}l(y)^xf_0(y)\mathrm{d}y\int_{\mathbb{R}}f_0(y)l(y)^x\ln (l(y)^x)\ln (l(y))\mathrm{d}y-\int_{\mathbb{R}}l(y)^xf_0(y)\ln (l(y))\mathrm{d}y\int_{\mathbb{R}}f_0(y)l(y)^x\ln (l(y)^x)\mathrm{d}y$$ is greater than $0$. Given: $\rightarrow f_0$ and $f_1$ are some density functions $\rightarrow l(y)=\frac{f_1(y)}{f_0(y)}$ is an increasing function. $\rightarrow x\in(0,1)$

The limit of hazard rate $h(x)=A/(1-B)$ as $x$ approaches $\pm \infty$

Can we tell what happens to the limit as $x$ approaches $\pm \infty$ of a hazard rate $h(x)$ defined for unspecified or generalized density as: $$ h(x)=A/(1-B) $$ where $A=f(x)$ is the density function, and $B=F(x)$ is the CDF.

Probability distribution of product of random variables using delta function

Suppose $A$ and $B$ are independent random variables, uniformly distributed on $[0,1]$. Define a new random variable $X = AB$. I would like to determine the probability density function $\bar{P}(x)$ for $X$. I know how to determine $\bar{P}(x)$ by computing the area under contours of constant $x$. The result is $$\bar{P}(x)=-\ln x.$$ However, I heard […]

Queue theory – M/D/k – Probability of never having a queue before a time T

This is probably a known result, but I couldn’t find any resource pointing directly to the issue I’m trying to solve. Suppose you start a logistic mission that needs that during its time $T_m$ a given object is always working. At the beginning of your mission, you have an instance of this object working plus […]

Distribution of sum of $n$ i.i.d. symmetric Pareto distributed random variables

Let $X$ be a random variable which follows the symmetric Pareto distribution. For a fix, real parameter set $\alpha > 0$ and $L>0$, its PDF is defined as $$ p_X(x) = \left\{ \begin{array}{ll} \frac{1}{2}\alpha L^\alpha |x|^{-\alpha-1} & |x| \geq L \\ 0 & \mbox{otherwise.} \end{array} \right. $$ If possible, I would like to derive PDF […]