Articles of probability

Why am I under-counting when calculating the probability of a full house?

I was trying to answer this question. Find the probability of getting a full house from a $52$ card deck. That is, find the probability of picking a pair of cards with the same rank (face value), and a triple with equal rank (different from the rank of the pair of course). My idea was […]

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\mathbf{x}[m] = [x_{m,1} \,\,\,\,\, x_{m,2} \,\,\,\,\, x_{m,3} \,\,\,\,\, … \,\,\,\,\, x_{m,N}]^\text{T}; \,\,\,\,\,\,\, 1 \leq m \leq M$ $\mathbf{y}[m] = [y_{m,1} \,\,\,\,\, y_{m,2} \,\,\,\,\, y_{m,3} \,\,\,\,\, … \,\,\,\,\, y_{m,N}]^\text{T}; \,\,\,\,\,\,\,\,\, […]

Expected time to roll all 1 through 6 on a die

What is the average number of times it would it take to roll a fair 6-sided die and get all numbers on the die? The order in which the numbers appear does not matter. I had this questions explained to me by a professor (not math professor), but it was not clear in the explanation. […]

Probability to obtain more than X with 3 dice.

This is a dice problem 1) I want to calculate the probability to have more than X throwing 3 dice of 6 faces. X = addition of the result of the 3 dice. 2) This is the first step but if you can also provide me a solution to calculate the probability to have more […]

Average distance between two random points in a square

A square with side $a$ is given. What is the average distance between two uniformly-distributed random points inside the square? For more general rectangle case, see here. The proof found there is fairly complex, and I looking for simpler proof for this special case See also line case

Thinning a Renewal Process – Poisson Generalization

If we have a Poisson point process with rate $\lambda$ and we keep each of its point with probability $p$, we obtain another Poisson point process with rate $\lambda p$. Does this result holds for a general Renewal point process? i.e., If in a renewal point process with rate $\lambda$, we keep each point with […]

Distribution of the sum of squared independent normal random variables.

The sum of squares of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then $X_1^2+X_2^2+\dots+X_k^2\sim\sigma^2\chi^2_k$. How do I go about obtaining the pdf? If I have $k$ independent normal random variables where $X_i\sim\mathcal{N}(0,\sigma_i^2)$ then what is the distribution of $X_1^2+X_2^2+\dots+X_k^2$?

how to derive the mean and variance of a Gaussian Random variable?

How do we go about deriving the values of mean and variance of a Gaussian Ransom Variable $X$ given its probability density function ?

Probablility of a dart landing closer to the center than the edge of a square dartboard?

The question is: There is a square dart board with a side length of $2$m, and a dart has equal probability to land anywhere on the board. What is the probability that the dart will land closer to the center than to the edges? So I know that if I center the square at the […]

The mode of the Poisson Distribution

Lately, I am doing an investigation on Stirling’s formula and its applications. So I thought I could use it to prove that the mode of the Poisson model is approximately equal to the mean. Of course, you do that by considering the curve that is formed by connecting the points of the probabilities of occurrence […]