This tag is for basic questions about probability and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity. For questions about the theoretical footing of probability (especially using measure theory), please ask under ...

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7 views

Maximum of independent Erlang random variables follows which extreme value distribution?

Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows gamma distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, ...
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0answers
17 views

Joint Expectation of independent Random Variables given two sigma-algebras

We have a question regarding two random variables $X$,$Y$ on a probability space with sigma-algebra $\mathcal{F}$ and a sub-sigma algebra $\mathcal{M}$ such that $X$ is independent of $\mathcal{M}$ ...
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2answers
12 views

Conditional probability of a Joint distribution

Let $(X,Y)$ have joint density $f(x,y)=e^{-y}$ , for $0<x<y$, and $f(x,y)=0$ elsewhere. What is $f_{X\mid Y} (x,y)$ for $0<x<y$? I think that the answer is $1/y$, however, I am having ...
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1answer
23 views

Odds of this happening [on hold]

The Edmonton Oilers have won the draft lottery 4 times in the last 6 years. These are their chances of winning each year. 48.1%, 48.1%, 18.8%, 4.7%, 14.2% and 11.5%. What is the percentage that ...
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2answers
17 views

Conditional probability of a poisson distribution

How could I calculate this conditional probability ${\large\mathsf P}\Big(N(5)-N(3)=6 \mid N(4) - N(3)=2\Big) $, N is a poisson process. Here is my try. $P(A|B) = \frac{P(A \cap B)(*)}{P(B)}$. but ...
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1answer
11 views

$A,B,C \in F $ A,B,C independent $P(ABC)=P(A)P(B)P(C)$ Prove: $A \Delta C$ and $B$ are independent.

$A,B,C \in F $ A,B,C independent $P(ABC)=P(A)P(B)P(C)$ Prove: $A \Delta C$ and $B$ are independent. I tried $$P((A \Delta C)B)=P(((A\C) \cup (C\A))B)=P((A\C)B ...
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1answer
17 views

Chance of Winning Sale [on hold]

If there is a 25% chance of winning a sale but I suck at selling so my chances of making this sale is 20%, what is my probability of winning this particular sale?
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1answer
14 views

How to compute Binomial Distribution?

A basketball player scores a point in a free throw with 80% probability. Probability is independent of the result of the previous throw. 1) Given 5 free throws, find the probability distribution of ...
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1answer
22 views

Question of Poisson Distribution

I'm still confused about how to apply the Poisson Distribution, could you help me to explain how to solve the following problem? A company department takes on average 2 new employees per year. New ...
2
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1answer
13 views

Independence of two random variables derived from a Brownian motion

If $X = B_1 + B_3 - B_2$ and $Y = B_1 - B_3 + B_2$ Where $B_t$ is Brownian Motion for $t \geq 0$ And I want to state with certainty whether $X$ and $Y$ are indep or not, do I simply just ...
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1answer
29 views

Distribution of Brownian Motion help

If $X = \frac{B_1 - B_3 + B_2}{\sqrt{2}}$ Where $B_t$ is brownian motion at time $t$. And I want to find the the distribution of $X$, how would I do so? $E[X] = 0$ is fairly straight forward. For ...
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0answers
33 views

EV of sequences of coin flips

I have a biased coin $C$ where $P(H) = p$ and $P(T) = 1-p$. This coin could be tossed an infinite number of times and the results will be recorded. I want to calculate the probability of a given ...
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2answers
21 views

when tossing a coin ten times, what is the probability of an outcome which has a string of 3 or more heads as well as a string of 3 or more tails?

here is an experiment from my Stat textbook "Try this experiment: Write down a sequence of heads and tails that you think imitates 10 tosses of a balanced coin. How long was the longest string ...
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0answers
11 views

Martingale difference sequence

Show that the sequence of random variables $w_t = (u^2_t - \sigma^2)x^2_{t-1}$ is a martingale difference sequence.
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1answer
18 views

How to extract a covariance matrix with this information

Referring to the above image, I wanted to know how to get the covariance matrix $\sum$. My understanding is, $A$, is our transformation matrice, such that $\begin{bmatrix} X_1 \\ X_2 \\ ...
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0answers
13 views

Help simplifying equation (sample variance is an unbiased estimator)

I would like to know how you get from this to $\sigma^2$. I tried to simply as much as I could and got
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0answers
8 views

Covariance and cross spectrum

A bivariate process $(x_t, y_t)$ is called stationary if each component is a univariate stationary process and $cov (x_s , y_{s+j}) =cov (x_t , y_{t+j}), \forall s,t,j$. The autocovariance function ...
2
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1answer
56 views

Let $X_1, X_2,\ldots$ be an infinite sequence of independent Bernoulli random variables

Let $X_1, X_2,\ldots$ be an infinite sequence of independent Bernoulli random variables where the probability of success is $1/3$ and the probability of failure is $2/3$. let $X$ be the number of ...
1
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1answer
23 views

Joint density function problem

I have a joint density function of Random Variables X and Y given by: $$ f(x,y) = \begin{cases} 2e^{-x}e^{-2y} & 0<x<\infty, 0<y<\infty \\ 0 &\text{otherwise} \end{cases} $$ And ...
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3answers
44 views

Calculating Var(X)

Let $\Theta$ be an unknown random variable with mean $1$ and variance $2$. Let $W$ be another unknown random variable with mean $3$ and variance $5$. $\Theta$ and $W$ are independent. Let: ...
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0answers
29 views

How do we improve the probability by repeating the experiment? [on hold]

In my class our sir told that probability of an experiment increases by repeating the experiment more times. for example i came through an example that Suppose a bin has white marbles and black ...
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1answer
24 views

Conditional probability - coin toss - getting 2 tails, then head in a row with unfair coins

We have four coins in a box, let $p_i$ describe the probability of getting head from coin $i$: $p_1=0, p_2=0.25, p_3=0.5, p_4=0.75$. We toss take one coin and toss it until we get head, what ...
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0answers
27 views

Probability and stats question [on hold]

Im kinda confused on how to approach this question and how to solve for it. i know there are multiple ways but i was wondering if anyone could help me. thanks 1.) It appears that the mean commuting ...
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1answer
22 views

Maesurability of function $\mathbb R\ni t \mapsto P(X>t)$

Let $(\Omega, S, P)$ be a probability space and $X$ be a measurable function on this space. Is a function $\mathbb R\ni t \mapsto P(X>t)$ Lebesgue or Borel measurable?
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0answers
41 views

How to randomly select a point from the surface of a unit sphere ?

Construct in $\Bbb R^k$ a random variable $X$ that is uniformly distributed over the surface of the unit sphere in the sense that $|X|=1$ and $UX$ has the same distribution as $X$ for orthogonal ...
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0answers
22 views

Why do we need to declare a probability measure for the definition of stochastic processes?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be measurable with respect to ...
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2answers
30 views

Finding average using central limit theorem.

I'm have a lot of trouble trying to wrap my head around how exactly to use the central limit theorem in this question. Image a gambling casino consisting of 100 busy roulette tables. Suppose that ...
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2answers
25 views

How to prove the inequality

For an arbitrary characteristic function $\phi(t)=\int_{-\infty}^{\infty}f(x)e^{itx}dx.$ Show that, $1-Re$$\phi(2t)\leq4\big[1-Re\phi(t)\big]$ and deduce that $1-\big|\phi(2t)\big|^2\leq ...
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0answers
13 views

Is the product of exponents of normal iid variables a martingale?

I am told that $X_1,\:X_2,\:,\dots$ is a sequence of i.i.d random variables, where $X_i\sim N(\mu,\sigma^2)$ for $i=1,2,\dots$ and that $Y_N=e^{X_1}e^{X_2}\dots e^{X_N}$. Is $Y_N$ a martingale?
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1answer
42 views

Trying to understand the behaviour of i.i.d.

In a course called introduction to probability theorem we are covering now i.i.d. (independent and identically distributed random variables). I already know when two variables are independent: $X, Y$ ...
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0answers
8 views

Does there exist a multivariate distribution on the positive quadrant that has tractable higher order moments?

In particular, let $X = (X_1, X_2)$ be a non-negative random vector. For what choice for the distribution of $X$, can I obtain tractable expressions for $\mathrm{E}[X_1^{r_1} X_2^{r_2}]$? I just need ...
7
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1answer
124 views

Expected area of triangle formed by three random points inside unit circle

Motivated by the discussion in The expected area of a triangle formed by three points randomly chosen from the unit square I tried to find an expression for the expected area of a triangle formed by ...
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0answers
14 views

When is a coupling ''natural''?

The definition of coupling is written below. In some articles, I found the term "natural coupling". When is a coupling said to be ''natural''? Definition of coupling between two random variables: Let ...
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1answer
24 views

Fifty men and thirty woman are lined up at random. How do I find the expected number of men who have a woman standing next to them.

Fi.fty men and thirty woman are lined up at random. How do I find the expected number of men who have a woman standing next to them.
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1answer
27 views

Solving for probability of dependent events

I was reading A First Course in Probability by Sheldon Ross. I read one of the problems and then tried building logic for it. Then read books solution which was completely different. So was guessing ...
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1answer
38 views

Simple probability with dice

Suppose you roll a 6-sided dice 6 times. a. What is the probability that all of the rolls show either 1, 2, or 3? Would the answer be $(1/2)^6$? b. What is the probability that all of the rolls ...
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3answers
31 views

Probability of at least one head given that there is at least one tail on three tosses of a fair coin

What is the probability of observing at least one head given that one observes at least one tail on three tosses of a fair coin. This is a poorly worded question in my opinion and am leaning ...
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1answer
23 views

What is the expected value of the highest of N independent draws from the unit uniform distribution?

I have a seemingly innocuous problem that I can't seem to wrap my head around. The following is mentioned in passing on one of my lecture slides, but when I try to arrive at the same conclusion I get ...
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1answer
44 views

What's more likely with 7 dice - Straight, Full House, or 3-of-a-Kind

Here's a problem I'm having. I have 7 dice. I want to rank the difficulty of achieving a particular 5 dice combination after all 7 dice have been rolled. A straight is when at least 5 of the seven ...
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1answer
18 views

small rate and probability

Is it true that when a rate is small, it can be considered as a probability of an event happening in a small time interval? For example, if a rate is Q [1/min], the probability of the event happening ...
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2answers
35 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
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0answers
22 views

p.d.f. of a position variable from stochastic velocity p.d.f.

I have a stochastic process, $v(t)$, that represents a velocity, and has a known probability distribution function $f(x,t)$ which is time-varying. I am interested to acquire a probability ...
2
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2answers
22 views

When does the probability of dealing 2 cards that I've already dealt in a previous shuffle reach 50%

I play poker and create poker software. So questions about card shuffling and dealing interest me. I'm trying to understand the likelihood of receiving the same hand twice over a number of shuffles. ...
2
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1answer
30 views

Drawing n intervals uniformly randomly, probability that at least one interval overlaps with all others

Randomly draw $n$ intervals from $[0,1]$, where each end point are selected from from the uniform distribution between $[0,1]$. What's the probability that at least one interval overlaps with all ...
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0answers
20 views

Product of (multivariate) Gaussian densities

One can frequently read that the product of two multivariate Gaussian pdf f1(x) and f2(x) is itself a Gaussian, with parameters as defined for example in: ...
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1answer
37 views

Two players toss a coin.

Two players $A$ and $B$ toss a coin. A has a coin $C_A$, B has a coin $C_B$. Probability of tail for $C_A = 1-a$, of head: $C_A = a$ Similary for $C_B$. Now, they are tossing on turn. The A starts. ...
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0answers
27 views

Does this sequence converge? If yes, what is the limit?

Assume $\{k_n\}_{n\geq 0}$ a sequence of natural numbers such that $k_0=0$, $k_n\leq k_{n+1}\leq k_n+1$, and $\lim_{n\rightarrow\infty} \frac{k_n}{n}=\alpha\in(0,1)$. So $\{k_n\}$ is an ...
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2answers
24 views

Find probability of event

Task is: Find probability of 4 aces laying in row in a deck of 36 cards. All possible shufflings of 36 deck is $36!$ I can place 4 cards in a row with $33$ different ways. And each way can be $4!$ ...
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1answer
21 views

Variance of a random variable between 0 and c.

My professor says we need to know how to solve a problem like this for our upcoming exam and I can't find anything in my textbook or notes related to this at all. Can anybody help make this ...
2
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1answer
15 views

Linear transforms of Normal dist [on hold]

If $X_t = \sqrt{t} Z$ where $Z \sim \text{N}(0,1)$ Then show the distribution of $X_t - X_s$ for $s<t$ Just wanted to check, would this be $\sim \text{N}(0,t-s)$ or $\sim \text{N}(0,(t-s)^2)$ ?