Probability Distribution

Let $\xi_n$ be a Poisson random variable with $\lambda = n \in \mathbb{N}$. That is $P(\xi_n = k) = \frac{n^ke^{-n}}{k!}$, for $k \in \mathbb{N}_0$. Let $f_+(n) = P(\xi_n \geq n),\ f_-(n) = P(\xi_n &g...

Firstly, as my assignment is due in 2 hours, I am offering $30. The due time for this post is set to 1 hour only - for the sake of speed, feel free to use pen/pencil on papeHi, I have solved the previ...

We have a d3 (sides 1, 2, and 3). We roll this die 15 times and record the frequencies that each side shows up. What is the EV of the product of these frequencies?For example, say we get 7 ones, 3 two...

Driver has driven 90,000 miles. And has completed 6,500 food deliveries(doordash, grubhub & ubereats). Driver has no history of accidents or car related infractions(tickets..). This is all the dat...

Hello!I'm trying to design a new dice system for a table-top roleplaying game. I think that a multinomial distribution or some sort of simulation (probably 100,000 simulations of rolling 10-sided dice...

Im in the middle of a research project and have all the raw data and calculated Mean and SEM values but am struggling to pick 3 correct t-tests to use (either paired t-test, 2 sample equal variance or...

We have a die with three sides (sides 1, 2, and 3). We roll this die 15 times and record the frequencies that each side shows up. What is the EV of the product of these frequencies?For example, say we...

Suppose we have a bin that has 6 balls in it. Five of the balls are blue and one of the balls is red. We pull a ball out, one at a time, and check to see if we've pulled out the red ball. If we have,...

Four independent random variables $X_i$, $1 \leq  i \leq 4$, are log-normally distributed such that $\ln(X_i)$ has mean $0.5$ and variance $1$. What is the probability that the product of the $X_i$ is...

What I already know: $P(x=k) = {{k-1} \choose{m-1}}*(1/2)^{k-1}$I suspect one way to get CDF is to Sum P(x=k) from k=m to x, and Mathematica evaluates that sum as:$ 2-2^{x}{{x} \choose{m-1}}*hypergeo...

Sailboat's route on map is 6km long. Because of the weather, the real traveled distance of the boat is random variable X (km) and speed Y (km/h). Density functions and sample spaces of these two indep...

Let $\vec{X}\sim N_d(\vec{\mu},\Sigma)$.  If $\vec{\mu}=\mu \vec{1}$ and $\Sigma=\sigma^2 I_d$, find the distribution of $\overline{X}_d$ conditioned to the event \{d\max\{X_1,\dots,X_d\}\in B\}, wher...

Two palyers battle in a competition and they each have 50% of winning 1 point at each round. The winner of the tournament is the first who is (a) two points ahead(b) three points ahead Let X be the nu...

Question on M/M/1 queuing see attached image

Recall the following definition of a Poisson process with rate λ > 0 : a counting process {N(t)} with N(0) = 0 is called a Poisson process with rate λ if the following are true: it has independent...

Not sure if I'm allowed to put more than one question in each, but lets go: A coffee machine fills a cup with a volume of coffee standard distributed with an expected 2.1dl and a standard deviation of...

I am attaching a picture of the question. thanks