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
11 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
2
votes
1answer
19 views

Finding the PDF from the CDF where the CDF is not differentiable at some point

I got the following problem: Let $X$ be a continuous random variable with $CDF$ denoted $F_X$ defined as follows: $F_X(x)= \begin{cases} 1-x^{-4/3}, & x\in[1,\infty) \\ 0, & x\in ...
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0answers
12 views

Find probability density function of random vector

Random vector has continuous distribution of setA={(u,v), v>=0, u+v<=1, v-u<=1}. I need to find joint probability density function of this vector. In my ...
0
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0answers
10 views

Limit distribution on return time $\tau = \inf\{k: X_k = X_m \text{ where }m<k\}$ [on hold]

Suppose there is a stochastic process ${X_i}_{i=1}^n$ where $X_i$ is distributed normally over $\{1,\dots,n\}$. As $n\rightarrow \infty$, the probability that any one value is repeated should go to ...
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0answers
44 views

Where can I find consultants on Long Lead Coin Tossing Experiments? [on hold]

Where can I find consultants on Long Lead Coin Tossing Experiments? I am specifically interested in the phenomenon of Long Leads, which was discussed in Feller's classic text. I want somebody to ...
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0answers
4 views

What is the formula for the 2-sample Anderson–Darling upper tail test?

There are computationally simple formulas for the Anderson–Darling test between an analytic distribution and an empirical distribution, as well as for the Anderson–Darling upper tail test (again ...
0
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0answers
23 views

The number of self-avoiding paths in the plane of length $k$

The number of self-avoiding paths in the plane of length $k$ is at most $4 \cdot 3^{k-1}$ according to this. Why? My immediate thought: four options for the first move and always three choices after ...
1
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2answers
25 views

Ladybug walking on a hexagon, mosquito walking on a number line probability question

1) A ladybug is walking at random on a hexagon. The ladybug begins at Vertex A. Each minute, the ladybug moves to ONE OF the randomly chosen TWO vertices adjacent to the one she's currently on. ...
1
vote
3answers
42 views

How many ways to make a connected graph using 4, 5, 6 edges?

How can/how many ways can you make a connected graph that has 5 vertices using 4, 5, 6 edges? I'm not sure how it would look like for 4 edges. Can you draw a diagram?
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2answers
41 views

Interchanging the order of a double infinite sum

I'm stuck at a proof of Wald's first equation about interchanging the order of a double infinite sum: Suppose $X_n \ge 0$ and $1_{\{\cdot \}}$ be indicator function. $$ \sum_{n=1}^\infty ...
0
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2answers
35 views

central limit theorem, solving for probability

I am playing a game and am trying to calculate the probability that I will win at 40,000 or more points total, if I play the game 1,000,000 times. The expected value for one game is zero and the ...
1
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1answer
16 views

Probability: Expectation: indicator RV, what is 1-((N-1)/N))^n?

Say there are N coupon types, you collect n coupons, and what's the expected number of types of coupons? My question is specifically about $1-(\frac{N-1}{N})^n$, the probability of getting a coupon ...
-1
votes
1answer
57 views

$3$ dimensional light up cube ornament, # of rows/cols/diags in/on a cube

Imagine a $3$ dimensional cube (much like a $4\times4\,(\times4)$ Rubik's cube) except the planes of the cube cannot be twisted individually and instead of faces with different colors, it is clear ...
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2answers
44 views

What is my probability space and measurable space?

I have the following difference equation $$ \tilde{u}_k = \begin{cases} u_k & \text{if $\gamma_k = 1$, no signal lost} \\ \tilde{u}_{k-1} & \text{if $\gamma_k = 0$, signal lost} \end{cases} ...
0
votes
1answer
34 views

Probability of the sum of exponentials being greater than a number

Say we have Xi ~ Exponential(1/3) We add Xi (all independent variables) until we reach a value of 5 or greater than 5 What is the probability of the sum of Xi being greater than 7? I have no idea how ...
0
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0answers
14 views

Cinlar Ex. 1.15: Trace space of a measurable space.

In constructing the trace space on a subset of a measurable space, it seems one has to assume that the subset is an element of the original measure space's sigma algebra, i.e., measurable in the ...
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0answers
21 views

AI Bayes Network Question? [duplicate]

A) Given this Bayes Net Answer and explain: 1) True or False 2) True or False B) Given this Bayes Net: Answer and explain: 3) True or False 4) True or False
1
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1answer
46 views

What is the probability of choosing r objects from c different groups when there are m groups of n objects?

Suppose I have m groups of n objects each for a total of nm objects. I am going to choose r of these nm objects. I want to know what the probability is that my r objects come from c different ...
1
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3answers
43 views

The probability of Breakeven On a Coin Toss Game

I was walking the other day around my work office in NYC and thought of this interesting scenario in a game of coin flips. You have $500 in your pocket. This is your entire life savings. You play a ...
1
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1answer
19 views

Four dice, probability that difference of some outcomes is equal to others

I roll four dice which gives me outcomes $x_1, ..., x_4$. I want to determine the probability $$P\left((x_2-x_1) = (x_4-x_3)\right)$$ I have already calculated other probabilities in this setting and ...
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0answers
28 views

AI Bayes network? [on hold]

A) Given this Bayes Net Answer and explain: 1) True or False 2) True or False B) Given this Bayes Net: Answer and explain: 3) True or False 4) True or False
0
votes
1answer
24 views

Determine the density of sum of three normal variables.

Setting $\pmb{X} = (X_1,X_2,X_3)$ is a properly center normal with covariance matrix $$\begin{pmatrix} a & b & 0\\ b & d & 0\\ 0 & 0 & e \end{pmatrix}$$ Determine the ...
1
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1answer
26 views

distinguishing probability measure, function, distribution

I have a bit trouble distinguishing the following concepts: probability measure probability function (with special cases probability mass function and probability density function) probability ...
0
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1answer
31 views

Probability that there is sub-sequence of exact length

Can you help me to solve the following: Find probability that in sequence of N random uniformly distributed numbers there is increasing sub-sequence of exact length L.
-1
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1answer
32 views

probability density function for random variables [on hold]

Suppose $x$ is a random variable with PDF $F(x)$ to be a continuous distribution. What is the probability of obtaining $x=a$ ?
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0answers
16 views

Standard Deviation, Random Sample, Probability Q #2 [on hold]

A. 90% of people eat their eggs with salt, 75% eat their eggs with pepper, and 65% of people put both pepper and salt on their eggs. What is the probability that a person eats his eggs with salt given ...
3
votes
1answer
22 views

Probability measures and stochastically dependent events

If $P(B\mid A) > P(B)$ and $P(C\mid B) > P(C)$ can I infer that $P(C\mid A) > P(C)$? My suspicion is yes but I don't see how to prove it yet.
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1answer
44 views

Standard Deviation, Random Sample, Probability

Suppose the average person spends \$16 per week on soft drinks, with a standard deviation of \$2.50. If a random sample of 47 people is taken, what is the probability that the mean amount spent on ...
0
votes
1answer
27 views

Constructing a joint distribution given $P(X\in A \mid Y)_\omega$

For random variables $X,Y,Z$, I am given for any measurable set $A$ $$P(X\in A \mid Y)=P(Z\in A\mid Y) \text{ a.s. }\iff (X,Y)\overset{d}{=} (Z,Y).$$ The direction $\Leftarrow$ doesn't seem too ...
2
votes
1answer
32 views

Consistency vs Inconsistency in a set of sentences: which is more common

I'm curious whether there is any research in the "probability" that a set of sentences in a first-order logic is consistent. Obviously, there are an infinite number of inconsistent sets and an ...
0
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3answers
23 views

probability of the empty set for arbitrary probability measures

I have a probability space $(\Omega, \mathcal{P}(\Omega), P)$. I want to know the probability of the empty set $\{\}$. Intuitively, I would say this probability is zero. It certainly is for the ...
2
votes
1answer
41 views

Is this an upper bound or lower bound?

I came across a probability distribution function in my work, it is however difficult to find in closed form, therefore I am looking to either upper bound or lower bound it. Assuming $a,b,T$ are ...
0
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0answers
15 views

determine how much probability increase with an added condition

Suppose there are $N$ people and $N$ prizes, and only $M$ out of $N$ are valuable. Every time one person is picked randomly, then he pick one prize randomly as well (this prize/person is then removed ...
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0answers
22 views

Joint distribution of arrival times in Poisson process

I need to compute the following joint distribution in a Poisson process: $f_{S_A S_{A+B}}(t_1, t_2), t_2\ge t_1$ $S_A$ and $S_{A+B}$ are the arrival epochs of the $A^{th}$ and ${A+B}^{th}$ arrivals ...
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0answers
8 views

Distribution of the ratio of two dependent chi-square

I look for my work the distribution of the ratio of two dependent chi-square variables $X, Y$ with different degrees of freedom for each one. Meanwhile I only found the distibution for the case where ...
2
votes
1answer
31 views

Uniform Distributions in Probability

X, Y, and Z are independent and uniformly distributed over [0,1]. I'm trying to find the distribution of XY by using the joint transformation T = X, W = XY. We haven't learned transformations yet, ...
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0answers
18 views

Determine if the following family of hash functions is universal

Let $H = \{h_1,h_2,h_3\}$ be the family of hash functions defined below, each mapping $\{a,b,c,d,e\}$ to $\{0,1,2\}$. Is $H$ universal? A family of hash functions is universal if $\forall ...
0
votes
1answer
9 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
0
votes
1answer
20 views

Approximate normal distribution(this is different from what I asked earlier $\log(n)$ is replaced by $\sqrt{\log{n}}$)

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
0
votes
1answer
21 views

Determine the probability distribution of a ratio of two random variables?

Setting You are given two independent random variables $X_0,X_1$ with common exponential density $f(x) = \alpha e^{-\alpha x}$. Let $R = \frac{X_o}{X_1}$. Determine $\Pr[R > t]$ for $t > 0$. I ...
0
votes
0answers
7 views

Convergence of third moment in central limit theorem

Previously, I asked a question here about the rate of convergence of expectations of absolute values to the expected value of a Gaussian. If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = ...
2
votes
2answers
43 views

The limit of an expected value vs expected value of a limit in this betting game

Setting The outcome $X$ of a slot machine takes values 1,2,or 3 with probability $p(1) = \frac{1}{2}$, $p(2) = \frac{1}{4}$, $p(3) = \frac{1}{4}$. We are given 3 for one odds, that is if we bet 1 ...
0
votes
1answer
29 views

Central limit theorem in the setting of Poisson variables

Setting Given $S_{\lambda} \overset{d}{\sim} \operatorname{Poisson}(\lambda)$. Let $G_{\lambda}(t)$ be the distribution function of $\frac{S_{\lambda}}{\lambda}$. I need to determine ...
1
vote
1answer
22 views

What can you conclude about the first moment of a variable given the 3rd moment exists and is finite

Suppose you are given a random variable $X$ and told that $E[X^3]$ exists and finite. Can you conclude that $E[X]$ exists and is finite? What about $E[X^2]$? How would you argue rigorously whether ...
1
vote
1answer
13 views

Using chi-square test for statistics with multiple options in one variable

i think that this is more mathematical question, i am doing some statistics on survey. There is questions with multiple choices, so for example, if there are 4 choices, i can pick 1st ,2nd and 4th. ...
1
vote
1answer
24 views

Random Sample vs Simple Random Sample

I am reading, just for fun, the book Essentials of Statististics of Mario Triola. I am trying to see the differences between Random Sample and Simple Random Sample. In the book I found these ...
0
votes
1answer
23 views

Probability distribution for putting balls in boxes in a correlated way

I'm looking for help finding a probability distribution: Right now I have a problem where I have N indistinguishable balls, which I need to put into K indistinguishable boxes, each of which can hold ...
1
vote
1answer
40 views

Using the geometric distribution to find the probability that between 4 and 6 devices will be tested

Quality control tests spark plugs until they find one that doesn't work. If the probability of a spark plug working is 0.99, what is the probability that they will test between 4 and 6 (inclusive) ...
0
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2answers
23 views

Are these transient or recurrent states in a Markov chain?

I have the following transition matrix for a Markov chain with states $A, B, C, D, E$ $ \left| \begin{array}{ccc} 0 & 0 & \frac{1}{2} & \frac{1}{2} & 0 \\ \frac{1}{2} & 0 & ...
8
votes
1answer
59 views

probability that no two spiders end up at the same vertex?

Eight spiders are located on the eight vertices of a cube. When a bell rings, each spider moves (at random, independent of the others) to an adjacent vertex. What is the probability that no two ...