Articles of probability

Using multivariate hypergeometric distribution to compute probability of multiple events

I am new here. I have researched this for over a month and spoken to 3 University professors and have not quite gotten the accuracy I am looking for. I am writing C# to compute probability based on user input and information in a database. I can calculate the probability of drawing at least x […]

marginal probability mass functions

Let $X$ and $Y$ be random variables with joint probability mass function $f(x,y) = k \cdot \dfrac {2^{x+y}}{x!y!} $, for $ x, y \in \{ 0, 1, 2, \cdots \} $ and for a positive constant $k$. Derive the marginal probability mass function of $X$. Also, Evaluate $k$. Are $X$ and $Y$ independent? Explain. Derive […]

100 coin flips, expect to see 7 heads in a row

So a random piece of information in a video I watched ages ago popped in my head tonight and I started thinking about it. I believe I am remembering this video properly… They flipped a coin 100 times you saw the ratio of head and tails to be 50/50. They created a diagram of all […]

Bounded random walk on one side only: Are you guaranteed to hit the bound?

Let’s say you have a one dimensional random walk, for example integers from 0 to infinity on the number line, and you start at some value $n$ with a probability $P(0)$ to take a step towards zero and a probability $P(\infty)$ to move towards infinity. My intuition tells me that no matter how small $P(0)$ […]

Expected value of $x^t\Sigma x$ for multivariate normal distribution $N(0,\Sigma)$

For standard normal distribution, the expected value of $x^2$ is $1$. A natural question is that in the multivariate case, what is the expected value of $x^t\Sigma x$ for multivariate normal distribution $x \sim N(0,\Sigma)$? I have difficulty to carry out the integral, but would guess the result is related to the norm of $\Sigma$.

Variance of event counting

I have this question (not homework, review problem for qualifying exam), tried approaching it a couple of ways (unsuccessfully). Any recommendations? Let $X_1,..,X_n$ be i.i.d continuous rvs. A record is said to occur at time $k$ if $X_k > X_i$ for all $i = 1,…,k-1$. Let $N$ denote the number of records. Find the variance […]

Why the two probability results are the same?

Suppose there are 3 red balls and 2 white balls in a bag. We want to pick out 2 balls without replacement. What’s the probability of the 1st and 2nd balls are both red? Solution 1: Use the conditional probability Let $E_1=$ 1st ball red $E_2=$ 2nd ball red $P(E_1)=3/5$ $P(E_2|E_1)=2/4$ then $P(E_1E_2)=P(E_2|E_1)\times P(E_1)=3/5 \times […]

Conditioning on a random variable

The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed between (0,5). That is Λ is uniformly distributed over (0,5), and given Λ = λ, the number of storms is Poisson with mean λ. Find the probability there are at least three storms this […]

Probability for having consecutive success in an experiment

A friend asked me the following question: “In an experiment, we are tossing a fair coin 200 times. We say that a coin flip was a success if it’s heads. What is the chance for having at least 6 consecutive successes?” And according to him, the answer is nearly 100%. My calculations were different. I’d […]

Probability of picked cards to be smaller than the largest picked card

I have an assignment for my algorithms module that requires us, amongst other things, to find the equations for the following question. Edit – Question Updated You have n cards with pairwise different integer values from 1 to n , shuffled randomly on a pile. You pick cards from the pile, one after another, and […]