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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2
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1answer
41 views
+50

Show $\int_{-\infty}^{\infty}\,f(u,t)dG(u)$ is a ch.f. where $G$ is a d.f. ; $f(u,\cdot)$ is a ch.f. and $f(\cdot,t)$ is continuous.

Show $$\int_{-\infty}^{\infty}\,f(u,t)dG(u)$$ is a ch.f. where $G$ is a d.f. ; and $f(u,\cdot)$ is a ch.f. for each $u$ and $f(\cdot,t)$is coutinuous for each $t$. Note that ch.f. means ...
0
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0answers
17 views

Continuous mapping theorem with density convergence

Let us consider a bivariate random variable $X\in \mathbb{R}^2$ with $pdf$ $f$. Also let, based on a sample of size $n$, let the the estimator of the density be $f_n(x)$ at $x\in \mathbb{R^2}$ and we ...
1
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1answer
17 views

Conditional probability of a zero inner product

Consider a random $n$ by $n$ matrix $M$ chosen uniformly over all $n$ by $n$ $(0,1)$ matrices and a random vector $v \in \{-1,0,1\}^n$ chosen uniformly as well. Let $X = Mv$. What is $$P(X_i = 0 ...
0
votes
1answer
17 views

Probability for rolling an odd number and tossing a coin on heads

A coin is tossed and a die rolled. Find the probability of getting a head and an odd number. The answer is $\frac{1}{4}$. My reasoning is that rolling an odd number is $\frac{1}{2}$, and tossing a ...
0
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1answer
10 views

Cumulant-Legendre

I have a short question: So suppose $b=\text{ess sup} X<\infty$, where $X$ is a random variable on $\mathbb{R}$. Now take $\Lambda (u)=\ln \mathbb{E}[e^{uX}]$, the cumulant, and ...
2
votes
4answers
72 views

A combinatorial proof for $\binom mk$+$\binom m{k-1}$=$\binom {m+1}k$

I do realize that there is a elementary proof of this result which follows from applying the formula $$\binom mk=\frac{m \cdot (m-1) \cdot \ldots \cdot (m-k+1)}{k!}.$$ I do wonder if there is an ...
2
votes
3answers
179 views

Stumped - How would I solve this probability question?

This question was merely a fun online math problem to see how many people could solve it, but I haven't been able to since last week and it's beginning to drive me nuts. The question: A man has $7$ ...
1
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1answer
48 views

An inequality about signed measure.

Suppose $\mu$ is signed measure,then: $$|\mu(A)|\le\epsilon\Rightarrow|\mu|(A)\le2\epsilon$$ I tried to use the Jordan composition of $\mu$: $$\mu^+(C)=\mu(C\cap D),\mu^-(C)=-\mu(C\cap D^c)$$ so ...
0
votes
2answers
18 views

how do they calculate these following columns

I have these data: I am sorry the data is in Portuguese, and it is an image so I can't convert it to a table but the translate "probably" ( i am not a native speakers for Portuguese language) is: ...
4
votes
2answers
34 views

Determining probability generating function for waiting time to see first $SS$

Given a sequence of Bernouilli trials, we have $P(S) = \frac{2}{3}$ with $0<p<1$. The event "SS" occurs on the $i$-th trial if we observe an $S$ on the $i$-th trial following a $S$ on the ...
1
vote
1answer
19 views

Comparing Percentiles of 2 Samples Drawn from the Same Distribution

Suppose I have two sets of numbers: $A=\{a_1,a_2,...a_{N_1}\}$ and $B=\{b_1,b_2,...b_{N_2}\}$ with $N_1<N_2$. WLOG assume that $a_i<a_j$ for all $i<j$ and similarly for $b_i$ and $b_j$. ...
1
vote
1answer
29 views

Combining Markov chains

If the following Markov chain relations hold: $$X \rightarrow Y \rightarrow Z,$$ $$Y \rightarrow W \rightarrow Z,$$ can we combine them to have $$X \rightarrow Y \rightarrow Z \rightarrow W ...
6
votes
1answer
44 views

Show rigorously that Pólya urn describes a martingale

We work with the famous Pólya urn problem. At the beginning one has $r$ red balls and $b$ blue ball in the urn. After each draw we add $t$ balls of the same color in the urn. $(X_n)_{n \in \mathbb ...
0
votes
4answers
71 views

mutually exclusive event vs independent event

Can you illustrate with examples, what is "mutual exclusive event" and what is "independent event". Without math equations, please elaborate it.. Thanks in advance
0
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0answers
12 views

Can anyone shed some light on the random variable which has the following characteristic function?

I have a random variable whose characteristic function is of the form \begin{equation} \mathbb{E}[e^{itX}] = \frac{(1-it)^a}{(1-2it)^{\frac{a}{2}}}\,, \end{equation} where $0<a<1$ I am not ...
1
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1answer
2k views

Independent Event Complements

I have the following homework assignment that I've already finished, but am confused on whether I've gotten right/wrong, and was hoping someone could help explain so I understand the problem better. ...
0
votes
1answer
10 views

Let X and Z form a random sample from a poisson dist.If Y=min( X,Z), what is P(Y=1)??

Let X and Z form a random sample of poisson distribution and define Y=min( X and Z) What is P(Y=1)?? I think Y is minimum of two. If X=1, then Z can be any number except 0 If Z=1, then X can be ...
0
votes
0answers
19 views

Calculate the probability in order to $f(y)=1$ provided $y\in \mathbb{R}^N$.

Given $x_1,x_2,...,x_n$ be $n$ points in $\mathbb{R}^N$.Let $f:\mathbb{R}^N\rightarrow \left\{ 0,1\right\} $. Suppose that $f(x_1)=f(x_2)=...=f(x_n)=1$. Calculate the probability in order to $f(y)=1$ ...
2
votes
1answer
32 views

Does the distribution of a process on $\mathbb{R}^{[0,\infty)}$ uniquely define it?

Question: Can I have two different stochastic processes $(A_t)_{t \in [0, \infty)}$, $(B_t)_{t \in [0, \infty)}$ having the same distribution on $\mathbb{R}^{[0, \infty)}$ differ in some ways? ...
0
votes
2answers
35 views

Dice Probability (increasing numbers)

If I have 6 regular dice, (each numbered 1-6): What is the probability that when rolled that each will be a different number.(each individual di is a different number from 1-6, but a random order) ...
1
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3answers
34 views

About discrete probabilities (Expected values)

Is my solution correct? Suppose two player (A and B) each one with 200,00 dollars toss a coin not balanced in a such way that the probability of head is $p$. Suppose yet that if the result obtained ...
0
votes
1answer
402 views

Best martingale for sequence of “dozen” bets at roulette game

Jim goes the Casino to play roulette. He only makes “dozen” bets at each spin ; his probability of winning is therefore $\frac{1}{3}$ every time (to simplify, we neglect the effect of the zeros in ...
0
votes
1answer
31 views

The probability of a number appearing in an approximation of an irrational number?

I was wondering if for the number Pi some numbers are more likely to appear than others, for example 3.141594 ... The number 1 appears twice does that mean that the probability for the number 1 ...
0
votes
1answer
14 views

How to calculate covariance of X and Y given joint probability

$X$ and $Y$ are dependent variables both normally distributed as $N(\mu-const, \sigma^2)$. I don't know what the joint distribution is. I know that when $const = 0$, then the joint probability ...
3
votes
1answer
147 views

The math behind generating Dungeons & Dragons ability scores: roll 4d6, toss lowest

D&D 5th ed. gives the following instructions for determining your “ability scores.” Roll four 6-sided dice and record the total of the highest three dice If I repeat the ...
1
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2answers
51 views

Given a Markov chain $X \rightarrow Y \rightarrow Z$, why is $I(X;Y|Z) \leq I(X;Y)$?

A Markov chain $X \rightarrow Y \rightarrow Z$ is given, where $X,Y,Z$ are random variables characterized by the probability distribution $p(x,y,z) = p(x)p(y|x)p(z|y)$. It follows that $I(X;Y) \geq ...
0
votes
1answer
22 views

Given probability of two elements being same in a list, find total number of unique elements

I have a list L, of numbers ordered randomly. Every number in the list is from a domain of $1$ to $100$ with the possibility of duplicates. If I point to(without removing) two numbers from the list ...
0
votes
1answer
22 views

Illegal lottery problem (Merging dependent bernoulli trials)

Suppose I am in a town that playing lottery is illegal. If I buy a ticket for 1 dollar, I will win the lottery with probability $p$. Each time I buy a ticket, the police may catch me and confiscate ...
1
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3answers
36 views

Normal distribution exercise!

If a technician does not encounters any hardware problems, the time he requires to assemble a computer follows a normal distribution with a mean of $30$ minutes and a standard deviation of $3$ ...
0
votes
1answer
44 views

Probability of a rolling a dice $n$ times with $k$ faces

I need help calculating the probability of rolling $n$ dice with $k$ faces. So you have multiple dice, all with $k$ faces (number of sides on a dice) and you want to calculate the probability of a ...
1
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1answer
18 views

Given Nd6, what is the probability that the two highest are minimum 4?

So, my statistics knowledge is rather poor, so I would welcome a formula explanation to the question: given Nd6 (6-sided dice) what is the probability that the two highest numbers are at least a 4? ...
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0answers
51 views

How many from 0 to 99999 [on hold]

How many times does the number 92789 appear in any sequence from 0 - 99999. If you know can you please include the formula.
0
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1answer
23 views

Markov Chains, reccurent and transient

Let the Markov Chain consisting of the states $0,1,2,3$ have the transition probability matrix ...
0
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0answers
17 views

Find the constants for the independence of a random variable

The following is my question: Let $W(t)$ be a standard Brownian motion, $\xi\sim N(\mu,\sigma^2)$, and $\xi$ is independent of $W(t)$ for all $t\geq0$. Define $X(t)=t\xi+\lambda W(t)$, for some ...
1
vote
4answers
49 views

Probability question involving infinite number of vertical chords in a 1 inch circle.

Infinite number of vertical chords drawn on a circle with a 1 inch radius. What is the probability that a randomly picked chord is shorter than the radius? The answer should be $1 - .5√ 3$ or ...
0
votes
1answer
465 views

Probability of dice roll (board games)

Assume that we have n * 6side dice. We will roll all n dice. I ask what is a probability of getting at least r * 1(number 1 on a die), s * 2, t * 3, u * 4? Number 6 can be used instead of any of other ...
0
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3answers
37 views

Golf Question when you have uneven players teams how can you make it simple and fair

We have various sizes groups sometimes they are even sometimes not. Most of the time we have foursomes equal groups sometimes we might have 3 foursomes one threesomes. We play golf we come into the ...
0
votes
1answer
43 views

An isosceles right triangle has legs of length 10. A pin is dropped into it and lands somewhere in the triangle where all places are equally likely.

What is the probability that it does not land within 2 units of any of the sides? From my calculations, I get that the smaller triangle has side lengths of 4,4, 4 root 2 (-2 at the right angle and ...
1
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0answers
41 views

Application of Slutsky's Theorem to the Convergence of Sum of R.V.

Let $X_1, X_2,…, X_n$ be i.i.d. $U(−\theta,\theta)$. Show that $Z_n \to N(0,\sqrt{5/9})$ in distribution, where $Z_n ...
2
votes
0answers
7 views

Likelihood that two markov chains are derived from the same transition matrix

Forgive me for my weak statistic background, hopefully what I'm asking makes sense. So some quick background, I have one markov chain from a data set and many additional chains that I'm producing from ...
1
vote
2answers
41 views

Distribution of Summation of two discrete random variables

Here, $\tilde{x}_1$ and $\tilde{x}_2$ are two non-negative independent discrete integer-valued random variable and the support set of $\tilde{x}_1$ and $\tilde{x}_2$ are below: $$ X_{1} = \{ ...
1
vote
1answer
32 views

Proving Properties of Discrete Time Markov Chain mathematically

I want to prove that the queue length at a store is not a Discrete Parameter Markov Chain (DPMC). Now I have the equation: $$Q_k = (Q_{k-1} - 1) + V_k$$ $Q_k$ is the queue length at time instant ...
-1
votes
0answers
18 views

Distribution of the maximum of correlated Gaussian variables

I try to unterstand how the distribution of the maximum of a iid Gaussian sample changes when we introduce correlation. For this, I simulated iid random samples and correlated samples with $\sigma =1$ ...
1
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3answers
51 views

Probability Of People Visiting Pubs

5 people went out to pubs. everyone individually choose a pub randomly from 10 pubs available. What is the probability that at "Pub1" "Pub2" came at least one person? $|\Omega|=10^5$ How can ...
1
vote
2answers
37 views

How to calculate $V(X+Y)$ with $X$,$Y$ dependent?

I want to calculate the covariance of two dependent variables $X$ and $Y$ and I don't know the value of $V(X+Y)$, that is, the variance of $X+Y$. I know how the quantities relate to each other: ...
0
votes
2answers
16 views

Probability of X given the sum

I am given that $X \sim P(\lambda)$, $Y \sim P(\gamma)$, and told to calculate the distribution of $P(X | X+Y = n)$ I proceed as follows $$ \begin{equation} \begin{split} P(X=i|X+Y=n) &= ...
3
votes
2answers
35 views

A grasshopper starts at the origin and is equally likely to hop north,s,e,w. What is the probability that it's coordinates will be 0,0 after 4 hops?

The grasshopper must hop in all $4$ directions (North, South, East, and West) to get back to the origin after $4$ hops. Therefore, I did: $\frac{(4 \cdot 3 \cdot 2 \cdot1)}{4^4} = .09375$. However, ...
-3
votes
1answer
19 views

prior probablity is lower than conditional probabilty [on hold]

Is there any reason behind the fact that prior probability is always lower than the conditional probability? I'm wondering why this happens..Can you please provide any reason for it?
-1
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0answers
15 views

Question concerning a proof on Stroock and Varadhan 1971

In the proof of theorem 2.3 of the article diffusion processes with boundary conditions (1971) one reads: where $Q_{s,x}$ is the unique solution to the martingale problem for $a,b$ starting from $x$ ...
0
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1answer
14 views

Expectation of size of bootstrapped sample

Lets say we have a sample $\mathbf{X} = \{x_1, x_2, \dots, x_N\}$. We draw $N$ points from $\mathbf{X}$ with replacement (do a $\textit{bootstrap})$. What is the expectation of size of bootstrapped ...