Articles of probability

Covariance between squared and exponential of Gaussian random variables

Assuming the random vector $[X \ \ Y]’$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]’$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X \ \rho \\ \sigma_Y \ \sigma_X \ \rho & \sigma_Y^2 \end{array} \right]$, I am looking for the expression of $\text{Cov}(X^2, \exp{Y})$. I know there […]

Check my answer: A slight modification to the 'hat-check' problem.

Suppose $n$ (hat wearing) people attended a meeting. Afterwards, everyone took a hat at random. On the way home, there is a probability $p$ that a person loses their hat (independent of whether other people did). What’s the probability that nobody got home with their own hat? First of all, I’m not sure if I […]

Relation between the two probability densities

Suppose $X_1, X_2, Y_1, Y_2$ are independent random variables on the same probability space with densities $f_1,f_2,g_1,g_2$ respectively. If $$ \int_{x} f_1(x)f_2(z-x) \,dx = \int_{y} g_1(y)g_2(z-y) \,dy \quad (*)$$ for all feasible $z$ and $$ \frac{f_i(x)}{g_i(x)} $$ is non-decreasing in $x$ for all $x$ in the support of $X_i$ and $Y_i$ for both $i\in\{1,2\}$ then […]

Are the events independent?

Three zinked coins are given. The probabilities for the head are $ \frac{2}{5} $, $ \frac{3}{5} $ and $ \frac{4}{5} $. A coin is randomly selected and then thrown twice. $ M_k $ denotes the event that the $ k $th coin was chosen with $ k = 1,2,3 $. $ K_j $ stands for […]

Joint distribution of dependent Bernoulli Random variables

I have $N$ Bernoulli random variables $X_1, …, X_{N}$ with known parameters $p_1, …, p_{N}$. I want generate a joint distribution in which these random variables are not independent as I know that joint distribution would just be the product of their marginals. How can I create this joint distribution that can be updated as […]

How common are probability distributions with a finite variance?

It’s always very surprising to learn that some of the entities one has been assiduously studying actually represent negligibly tiny minorities (e.g. continuous functions vis-à-vis all functions)… Now, the Central Limit Theorem, for one, holds only for probability distributions with a finite variance. How common are such distributions in the space of all probability distributions? […]

Find the probability density function of $Y=3X+2$.

A continuous random variable $X$ has probability density function, $$f(x)=\begin{cases}\frac{3}{5}e^{-3x/5}, & x>0 \\ 0, & x\leq0\end{cases}$$ Then find the probability density function of $Y=3X+2$. So I have difficulty in solving this problem. I am confuse about how to relate the density function of $X$ to that of $Y$. So I look up for hints, so […]

“Probability” of a map being surjective

What is the probability of randomly putting $n$ elements into $k$ boxes ($k\leq n$) such that there is no empty box? I have two different ideas: I could use the principle of inclusions and exclusions with $A_i=\{\text{box $i$ is empty}\}$: \begin{align}P(\text{no empty boxes})&=1-P\left(\bigcup_{i=1}^k A_i\right)=1-\sum_{i=1}^{k-1} (-1)^{k-1} \binom{k}{i}\left(\frac{k-i}{k}\right)^n\\&=\sum_{i=0}^{k-1}(-1)^k\binom{k}{i}\left(\frac{k-i}{k}\right)^n\end{align} There are $\binom{n+k-1}{k-1}$ different ways of putting $n$ elements […]

What is the pmf of rolling a die until obtaining three consecutive $6$s?

I am trying to find the pmf of rolling a die until 3 consecutive 6s turn up. I am able to find the expected value using a tree diagram, but the pmf is not obvious to me. Let A be the event of not rolling 6, and let B be the event of rolling a […]

Calculating probability with n choose k

I’m trying to solve a problem two different ways, and I can’d seem to figure out where I’m going wrong. I have 4 buckets (A,B,C,D), and 4 identical marbles. Each marble has an equal chance of being put in any of the 4 buckets, and each is placed independently (each bucket can have 0-4 marbles […]