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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Compute the stationary distribution of a Markov Chain on an infinite state space

I have a Markov Chain on $\mathbb N_0^2$ with a given initial state $(x_0,y_0)$. The allowed transitions for example are of the following form: $(x,y) \mapsto (x-1,y+2)$ with probability $\propto x$ ...
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0answers
12 views

mathematics of repeated carry overs

I want to know distribution of residuals material in subsequent refills. Say, a cup used for transferring salt is used without cleaning for transferring floor (say 10 times). Obviously, First transfer ...
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1answer
25 views

It is true that $\int_{0}^{\infty}\mathbb{P}(x<m \ \cap Y \leq k-x) f_{X}(x)dx= \int_{0}^{m}\mathbb{P}( Y \leq k-x) f_{X}(x)dx$?

Let $X$ and $Y$ be independent random variables. Then it is true that? $$\int_{0}^{\infty}\mathbb{P}(x<m \ \cap Y \leq k-x) f_{X}(x)dx= \int_{0}^{m}\mathbb{P}( Y \leq k-x) f_{X}(x)dx$$ And, how ...
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0answers
4 views

How to get an approximation of $P(A\leq \max_{1\leq i\leq n}B_i)$,where $A, B_i$ are independent Gaussian random variables

Consider the independent Gaussian random variables, $A$, $B_1$,...,$B_n$. $B_i$ is distributed as $N(0,1)$. They are all independent. $A$ is distributed as $N(m,1)$. How can I approximate the ...
3
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3answers
208 views

Probability of drawing a pair of brown socks

You have a drawer with $6$ loose blue socks, and $10$ loose brown socks. If you grab two socks from the drawer in the dark (random draw), what is the probability that you draw a brown pair? I have ...
2
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1answer
111 views
+50

Result of a $2D$ random walk with position dependent probabilities

I was just wondering about $2D$ random walks when I got the idea of a position dependent $2D$ random walk: A man is initially at $(x,y)$ and can move in a line parallel to the X and Y-axis only. ...
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0answers
15 views

Bayesian statistics and Basis for continous functions

I was thinking about Bayesian statistics, and one thought bothered me: In Bayesian statistics, we assume that the pdf $p(x)$ can be described as: $p(x)=\int f(x|\theta)g(\theta)d\theta$ usually ...
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2answers
28 views

Finding the mode of a distribution

I've been trying to get a better understanding of distributions. So far I understand how we get the formulas for mean and variance (by looking at the derivative of the moment generating function). ...
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0answers
55 views
+50

Probabilistic interpretation for representation of unity using the zeta function

There's a cute identity, I believe due to Borwein, Bradley and Crandall (Section 4): $$1=\sum_{n=2}^\infty (\zeta(n)-1).$$ There are some generalizations in the linked paper as well. Question: Is ...
2
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2answers
39 views

Expected value - product of functions of uniformly distributed variables

We have $n$ random variables $X_1,...,X_n$, $n\geq 2$, where $X_i∼U(0,1)$ and all of them are iid. Let $ Z=\min(X_1,...,X_n)$ and $ \overline{X} = \frac{1}{n}\sum_{i=1}^{n}{X_i}$. Calculate ...
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1answer
18 views

Help needed with Probability Question

A card is drawn at random from a deck of playing cards. If it is red, the player wins 1 dollar; if it is black, the player loses 2 dollars. Find the expected value of the game.
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0answers
30 views

Separability of the Wasserstein space with respect to $W_2(\cdot,.) +|\phi(\cdot) - \phi(.)|$

I would be thankful, if someone could give me some short proof or reference for the following problem. Given a lower semi-continuous and geodesically convex functional $\phi$ on the Wasserstein ...
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2answers
57 views

Probability of each person writing code--in a certain language

I am little lost with this problem. Not sure which formulas to use A project was implemented by three developers: Pat, Jon, and Maria. They used four languages: C, C++, Python, and JavaScript. The ...
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1answer
37 views

Rolling 1 die 5 times

One die is rolled five times. How many different results are possible? Of those, in how many ways can there be exactly 2 rolls of 4?
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1answer
22 views

Decomposing factorized entropy

I am trying to figure out how the equation for factorized entropy below is derived. The equation for entropy is $H(Q) = -\sum_x Q(x)\log Q(x)$ where $Q$ is a probability distribution. Let $Q(x) = ...
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1answer
11 views

p-average compound metric

I'm trying to prove that probability space metric defined as $d(X,Y)=(\mathbb{E}|X-Y|^p)^{1/p}$ is a metric indeed. Literature states that $d(X,Y)=0$ implies $Pr(X=Y)=1$, but no further explanations ...
2
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1answer
48 views

Coin tossing: Streak count

I have a special request with regards to probability. Let's say I toss a coin 400 times. What I need to know is the average number of streaks of various lengths within such a sample. How many ...
2
votes
1answer
14 views

Conditional probability with max(X, Y)

Let $Y_n=$ the outcome of the $n$-th die roll, let $X_{n+1} = \max \{X_n, Y_{n+1}\}$ with $X_1=Y_1$. What is $P(X_{n+1}=j \ | X_1=i_1, ..., X_n=i)$? I know that it is $P(\max \{X_n, Y_{n+1} \}=j \ | ...
4
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1answer
76 views

Coin Flips and Hypothesis Tests

Here's a problem I thought of that I don't know how to approach: You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with ...
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0answers
11 views

Suggestions for dealing with these order statistics

Consider a collection of $n$ random variables $X_i \sim N(\mu, \sigma^2)$, ($i = 1,2,\ldots, n$) and a random variable $X \sim \text{Exp}(\lambda)$. All $X_i$'s and $X$ are mutually independent. Let ...
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0answers
11 views

multivariate interval estimation

I have several samples of probabilistic vectors, i.e, each sample is of the form $(x_1, \cdots, x_n)$ such that $\sum_{i=1}^n x_i\leq 1$ (they are sub-probabilistic vectors), how can I obtain a ...
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0answers
43 views

How to compute P(|X - E_Y[h(y)]| < c)?

Consider the discrete random variable $Y$, the continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, ...
2
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4answers
190 views

Median $\neq$ expectation

Do you have an example of real random variable such that its median is remarkably different from its expectation? I'd like an example where it is obvious that they are different.
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1answer
46 views

Question about Measure Theory [on hold]

Let $(\Omega, U, P)$ be a measure space and X be random variable and its distribution function $F_x(x)=P(\{\omega: X(\omega)\le x\})=P(-\infty , x]$ and the function F is continuous at x. If the ...
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1answer
26 views

Proving a statement about probability theory

Let X be arandom variable. Consider any constant $c\gt 0$ how to prove the following satement $$\sum P(|X|\ge cn) \lt \infty \iff E(|X|)\lt \infty $$ My answer trail: $E[|X|]=\sum_X|X|P_x(X)\lt ...
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1answer
20 views

Calculating the probability of getting a full bucket in a hash table with open addressing

I have a problem where I'm trying to calculate the probability of getting a full bucket when I use a hash table with open addressing. What I have: A hash table with 128 buckets, each bucket can ...
0
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1answer
36 views

Understanding the sum of random variables

I am currently learning probability theory. I have two questions: I would like to know through an example what is meant by the sum of random variables (r.v.). To make things simple let consider only ...
0
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1answer
38 views

some properties of $\nu$ measure

For any given function $F$ satisfying the following properties $0\le F(x)\le1,\forall x\in\mathbb R$ $F(x)\le F(y),x\le y$ $\lim_{x\to-\infty}F(x)=0,\lim_{x\to\infty}F(x)=1$ $F$ is continuous from ...
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2answers
22 views

What is the probability of winning in a shootout? [on hold]

Person A can make $\frac{2}{5}$ of his free throws Person B can make $\frac{3}{4}$ of his free throws They take turns with person A going first The first person to make his free throw is the winner ...
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1answer
27 views

probability for beginners : simple question

Could you answer the following please... If we roll a die once and define Event A: The face value is even but less than 6 Event B: The face value is not 1 or 6. a) Then what is the ...
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1answer
26 views

Bayes theorem - is it applicable in any case?

I'm studying the Bayes' Theorem and I have a doubt. In this wikipedia page there's an example of application for the following events: ...
0
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1answer
27 views

Simultaneous density function of two continuous variables, X and Y.

I'm having issues with calculating the simultaneous density function of two continuous variables, X and Y. I took a screenshot of the information: How should I start? I know that if the two ...
1
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1answer
35 views

interpreting wording of probability question

Two dice are rolled, and the sum of the face values is six. What is the probability that at least one ofnthe dice came up a three? I want to make sure that I am interpreting the language right when ...
2
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0answers
25 views

Hoeffding’s inequality extension

In Hoeffding’s inequality we assume that the random variables $X_i$ ,$i=1,..,n$ are i.i.d. and bounded . Is there any extension to Hoeffding’s inequality for the case that $X_i$ are identically ...
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1answer
109 views

Is it possible to calculate/impute the sample size “N” from a given mean and standard deviation?

Can anyone provide with some advice? - thank you I am required to calculate the sample size for two groups, given the following data: Total sample N (group1+group2)=583 group 1: mean=8.35, SD1.07, ...
2
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1answer
15 views

How to find data distribution law using MATLAB?

Having a random variable $T \geq 0$ and a set of discrete data represented by $t=t_i$ and $P(T \leq t-i)$. My aim is to find the distribution law of $T$. Is there any fast method in Matlab that can ...
0
votes
2answers
36 views

Let X have density 2t on 0 < t < 1 and Y be uniform on the interval (0, 10) and independent of X. Find the density of Y/X. [on hold]

Let X have density 2t on 0 < t < 1 and Y be uniform on the interval (0, 10) and independent of X. Find the density of Y/X. Find E(Y/X) I have no ideas how to solve it now i ...
2
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0answers
21 views

Find the correct combination

Case 1 : if we bet on team1 with Rs.1 and win then we will get Rs.1+Rs.1 Case 2: if we bet on team2 with Rs.1 and win then we will get Rs.1+Rs.3 Case 3:if we bet on team3 with Rs.1 and win then we ...
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3answers
243 views

What is the probability that 5 randomly chosen cards in a deck add up to 40 or greater?

I have made a probability game, where you have to pull out any 5 cards without looking (from a deck of 52 cards), and if all five cards add up to 40 or more, they player pulling the 5 cards from the ...
2
votes
1answer
20 views

Derive probability mass function from probability-generating function

Given the probability generating function $$G(z) = \frac{1}{2} \frac{3+z}{3-z}$$, how can one derive the pmf? I know that I have the manipulate the function into a series: $$G(z) = ...
1
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1answer
29 views

Steve Nash’s expected value from his one-and-one free throw situation is 1.72 points. What is his free-throw percentage?

The one-on-one free throw situation works like this - for the first throw, if you make it, you get to do it again. If you miss, you don't get another chance. If you make it the second time, you get ...
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1answer
32 views

Understanding summations with Poisson

I'm currently doing a problem on Poisson processes and I've encountered the situation where I'm not sure why this summation is expanded as follows: And similarly I have tried expanding out the ...
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0answers
25 views

calculate a probability using the central limit theorem

$X$ is a variable of a Bernoulli distribution $ X \sim b(p)$ where $p\in(0,1)$. We also have the sequence of independent and identically distributed variables $Y_n$ with uniform distribution. $ ...
2
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0answers
25 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
1
vote
2answers
34 views

Why $p\{N>n\}=p\{X_1+…+X_n\leq x\}$.

Let $(X_k)$ a sequence iid of random variable uniform on $[0,1]$. Let $x\in]0,1[$ and $N=\min\{n\geq 1\mid X_1+...+X_n>x\}$. Why $$p\{N>n\}=p\{X_1+...+X_n\leq x\} \ \ ?$$
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0answers
12 views

bias reduction when the bias depends on the true parameter

Let's say we estimate a parameter, $\theta$, by $\hat{\theta}$. For this estimator we have the following property that $$\hat{\theta}\to_{p}\theta+f(\theta)$$ where $\to_{p}$ denotes convergence in ...
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1answer
30 views

probability question of balls [on hold]

what is the chance of getting at least one defective item if 3 items are drawn randomly from a lot containing 6 items of which 2 are defective?
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2answers
42 views

Martingale definition

To prove that one process is Martingale, generally we prove 3 things : 1) X is adapted. 2)$$ \mathbf{E} ( \vert X_n \vert )< \infty $$ 3) $$\mathbf{E} (X_{n+1}\mid X_1,\ldots,X_n)=X_n $$ I ...
0
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1answer
54 views

Maximum likelihood estimator and confidence interval

Let $\theta$ be an unknown constant. Let $W_1,…,W_n$ be independent exponential random variables each with parameter $1$. Let $X_i=θ+W_i$. First, I need to find $\hat\theta _{ML}(x_1,\ldots ,x_ n)$. ...
6
votes
2answers
129 views

Rolling two dice, what is the probability that two consecutive $7$s happens earlier than a $12$?

Alice and Bob are playing a game involving two dice. If a sum of 12 appears, Alice wins and they stop playing. If a 7 appears twice in a row, Bob wins and they stop playing. What is the probability ...