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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1answer
20 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
15 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
16 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
61 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 ...
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1answer
26 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
46 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
32 views

How to understand the definition of weak convergence of stochastic processes

I have some problems with the definition of $\textit{weak convergence of stochastic processes}$. To ask my question, we start with two well-known definitions corresponding to measures and random ...
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1answer
28 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
33 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
49 views

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

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
27 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
36 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 ...
2
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2answers
29 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
16 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
52 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
10 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 ...
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2answers
164 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
17 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
35 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
30 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
58 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
37 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
35 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
48 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
19 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
36 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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1answer
33 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
27 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
37 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
23 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
38 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
30 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
23 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
21 views

Linear transforms of Normal dist [closed]

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 $\sim \text{N}(0,(t-s)^2)$ ?
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0answers
33 views

Guessing a password from a list of N passwords

I want to know if my solution to the problem is correct: I am given a list of n passwords to enter an account and only one will grant me access to it. I pick one and I test it. If it's incorrect, I ...
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1answer
30 views

Assume that P(B) > 0. Prove that if P(A1|B) < P(A1) then P(Ai | B) > P(Ai) for some i = 2, … , k.

Suppose k events from a partition of the sample space Ω, i.e., they are disjoint and ∪ i=1 to k over Ai = Ω. Assume that P(B) > 0. Prove that if P(A1|B) < P(A1) then P(Ai | B) > P(Ai) for some i = ...
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0answers
31 views

probability of collision approximation,matlab,maple

I'm working in the field of communication network. I have this equation p=((1-(1-(((2.*(1-2.p))./(1-p-p.(2.p).^m)).(1./Wi))).^(n-1))) I want to solve for p, but I failed(maple and matlab), can ...
1
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1answer
13 views

Asymptotic stopping time for a ball-drawing problem

Take two different boxes, one with $N$ red balls and one with $N$ blue balls. Remove balls one at a time from either box with equal probability. When only one color is left, the (expected value of ...
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2answers
19 views

Two processes. Expected value.

On a computer running two processes $ X_1, X_2 $ at the same moment. $ X_1, X_2$ mean time work processes, respectively. $ X_1, X_2 $ have exponential distribution. $$ E (X_1) = E (X_2) = 60s.$$ ...
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2answers
85 views

Clever way of finding $\int_0^\infty x\Phi(x)\phi(x)dx$

Suppose that $\Phi$ and $\phi$ are the Standard Normal c.d.f and p.d.f. respectively. Then, evaluate $$\int_0^\infty x\Phi(x)\phi(x)dx$$ There is no use of my trying to show my approach because ...
1
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1answer
15 views

Construction of Probability Generating Function in Branching Process?

So I'm trying to construct a probability generating function for the following scenario: 1/5 of a rabbit population does not reproduce. 4/5 have 3 offspring each, and the probability of male or ...
0
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1answer
40 views

Counting and Probability String Length

Consider strings that can be made up from the set $\{a, b, c, d, e, f, \cdots, z, 0, 1, 2, \cdots, 9\}$ 1) How many strings of length 8 contain either the letter '$x$' or '$1$'? 2) What is the ...
1
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1answer
22 views

Probability of number of drawing cards in a scenario being equal to that in another scenario

I came across the following question in a book:- $Q.$ Cards are drawn one by one at random from a well shuffled pack of $52$ cards. $(a)$Find the probability that exactly $n$ cards are drawn before ...
1
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1answer
24 views

Proving the sequence $f_{x_n}(x)= (n+1)x^n$ converges in distribution

I am preparing for a final exam and just working on sample problems. Let $X_1,X_2,\dots$ be an infinite sequence of continuous random variables such that $f_{x_n}(x)= (n+1)x^n$ for $0<x<1$ ...
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0answers
14 views

Probabilities for the repetition of the same experiment $N$ times

Sometimes one experiment we want to discuss in terms of probabilities is in truth the same as performing another experiment $N$ times. I have a doubt on how to relate the probabilities for the ...
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2answers
41 views

What is the expected number of people who are shorter than both of their immediate neighbors?

A total of n people randomly take their seats around a circular table with n chairs. No two people have the same height. What is the expected number of people who are shorter than both of their ...
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1answer
22 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...