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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Uniform Distribution: finding the probability between two variables

Q: In a uniform density $\mathcal{U}(a,b)$ with $a=-0.025$ and $b=0.025$, what is the probability that an error will be between 0.010 and 0.015? A: From the density function, I didn't know how $d$ ...
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2answers
96 views

Conditional events that are not in the event algebra?

The Wikip. page on conditional event algebra states that: David Lewis showed that in orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any ...
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1answer
57 views

What is the distribution of a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$?

From Did's comment following his reply, given a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$ for some function $r:[0,\infty) \to [0, \infty)$ every $t⩾0$. Is there a name for such a ...
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3answers
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Dealt 3 cards. Odds of being dealt any pair?

This is not to aid a gambling habit. I am simply curious how to do this math. You get dealt 3 cards. What are the odds of having any pair? (We can exclude 3 of a kind) Total number of hands = ...
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1answer
77 views

Finding $EX$ of a density function (integrating $\ln u$ over infinity)

I've been given a density function as: $f(x) = 1/4e^{-|x|/2}$ where $-\infty < x < \infty$ and need to show that $EX = 0$ I understand that to find $EX$ I must calculate $\int xf(x)~dx$ ...
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0answers
91 views

Relation between transition rate and hazard function?

For a stochastic process $X$ with continuous time and discrete state space, if $\forall i$ in the state space $$ P(X_{t+h}=i \mid X_t=i) = 1 - r_{ii} h + o(h) $$ and for $\forall j \neq i$, $$ ...
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2answers
1k views

Probability of a picking a white ball in second draw

Qn:Two balls are selected sequentially from an urn containing six red, three white, and four blue balls. What is the probability of selecting a white ball on the second draw if the first ball is not ...
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59 views

Need advice: what should be my next step?($2$) (does Cauchy-Schwarz help here?)

This question is based on the question that I asked here Need advice: what should be my next step? I did a little more progress and wanted to share with you. As this is a new question, without any ...
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0answers
61 views

Is this stochastic process called kernel?

I am reading a paper and spot the following concept. Given a sequence of random probability $p_n \in [0,1], n \in \mathbb N$ of some events, assume there exists a stochastic process $X: \Omega \times ...
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1answer
573 views

Rayleigh distribution

I have this question from my statistical theory course: A sniper shoots at a target. X and Y measure its deviation on the x and y axes. X and Y are independent and are distibuted normally with mean=0 ...
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3answers
89 views

a distribution function

If there are two multivariate independent gaussian variables, with their distribution function $F_1$ and $F_2$ then by what conditions the function $F(x):=F_1(x)+F_2(x)-F_1(x)F_2(x)$ is a ...
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2answers
53 views

The number of positive integers

The number of positive integers $abcd$ that have four digits such that $d\not=0$ and $a\not=0$ and number $4$ divide both $abcd$ and $dbca$ ? My attempt:
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2answers
451 views

joint probability of dependent variables

Consider three integer random variables $a, b$ and $c$ with common distribution function, $m(-1)=m(0)=m(1)=\displaystyle\frac{1}{3}$. What is the analytic way of calculating $P(ab=1, ac=-1)$ ? Many ...
2
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1answer
39 views

The number of sequences

The number of sequences that have five letters $A$ and three letters $B$ and two letters $C$ such that the first appearance of the letter $A$ is before the first appearance of the letter $B$ ? My ...
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1answer
286 views

How to determine the expected value of the $f(x,y)$?

How to determine the expected value of the $f(x,y)$ defined as: f(x,y): $\quad$ for i = 1 to y $\quad$ $\quad$ do x = R(x) $\quad$ $\quad$ return x where $R(N)$ returns any ...
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1answer
185 views

Markov chain stochastic process

Can anyone help me with this question, maybe by giving a hint. Consider a Markov chain with state space $\{0,1,2....\}$. A sequence of positive numbers $p_1,p_2,...$ is given with $\sum p_i=1$. ...
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1answer
150 views

Estimating the number of tickets bought in a lottery

A national lottery has the format where $7$ numbers are chosen from $45$ without replacement. The first $6$ numbers chosen constitute the "winning numbers", while the last number chosen is the ...
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1answer
143 views

Expression for the size of type class, or multinomial coefficient.

The notations follow those in Cover&Thomas, "Elements of Information Theory", 2ed. I saw from a paper that the size of type class $T(P)$ can be expressed as ...
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1answer
334 views

Repeatedly Toss Balls into Bins

$n$ balls are randomly tossed into $m$ bins, each bin can hold $k$ balls. If a ball is tossed into a full bin (already has $k$ balls in it), it can be tossed repeatedly and randomly into the $m$ bins ...
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1answer
48 views

Bird's Probabilities!!!

Please, could somebody help me to figure this exercise? You collect data on the wing lengths (Yi) of 50 bald eagles in one sample. The average wing length (Y sample average) is 89 cm and the ...
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1answer
487 views

What is the optimal strategy for this game?

You are playing a game where you put in a certain amount of money $m$. A random number in $[0, 1]$ is chosen. If the number is greater than $p$, you now have k% more money, otherwise, you lose all ...
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1answer
2k views

Conditional Probability with balls in an urn

Two balls, each equally likely to be colored either red or blue, are put in an urn. At each stage one of the balls is randomly chosen, its color is noted, and it is then returned to the urn. If the ...
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0answers
197 views

Probability Tree with three events at the same time - Help!

I'm not sure if the answers are right. I wonder if somebody could help me? Thanks Consider a population of fish (of the same species) that have different color patterns. 30% of the fish are green, ...
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1answer
71 views

Conditions that imply Lindeberg's condition

Suppose that $Z_{1},Z_{2},\ldots$ are i.i.d. random variables with mean 0 and variance 1, and define $X_{nk}=\sigma_{nk}\cdot Z_{k}$ If $$ \frac{\underset{1\leq k\leq ...
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2answers
5k views

Probability of at least two events occurring.

The proportion of the American adult population that supports candidate Green is p=0.22. A SRS of 9 adults asks if they agree with the statement “I support candidate Green.” What is the probability ...
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1answer
381 views

CDF of $X+Y$,$X−Y$,$XY$ for $(X,Y)$ Chosen Uniformly Inside Triangle

Let $(X,Y)$ be chosen uniformly on the triangle $\{(x,y)\in\mathbb R^2:x+y\leq1,x\geq0,y\geq0\}$. What is the joint density function of $(X,Y)$? Find the CDFs of $X+Y$, $X-Y$, and $XY$. What I've ...
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1answer
78 views

Probability of “at least one event”.

The proportion of the American population that has disease $Z$ is $p=0.02$. If $55$ people are randomly selected from the population, what is the probability that at least $1$ of them has disease ...
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1answer
2k views

Erroneous Answer Key?

The problem I am working on is: The current in a certain circuit as measured by an ammeter is a continuous random variable $X$ with the following density function: $f(x) = .075x + .2$ for $3 \le x ...
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0answers
169 views

Need advice: what should be my next step?

I am dealing with a quite algebraic question and I arrived at some good point. I had $2$ equations with $2$ unknowns and I was able to eleminate one of the variables. My final equation still seems ...
2
votes
2answers
365 views

Normal approximation to the binomial distribution

As read on Wikipedia, the binomial distribution $B(n, p)$ is approximately normal with mean $np$ and variance $np(1−p)$ for large $n$ and for $p$ not too close to zero or one. Why ? Why this condition ...
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3answers
469 views

How many papers do you expect to hand in before you receive each possible grade at least once? [duplicate]

A particular professor is known for his arbitrary grading policies. Each paper receives a grade from the set {A, A-, B+, B, B-, C+}, with equal probability, independently of other papers. How many ...
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2answers
610 views

How to create a transition matrix that will guarantee an outcome after infinite transitions

Let's assume we have the a transition matrix like: 0 0 0 1 2 0 2 4 0 3 6 0 4 7 2 5 9 3 6 6 6 7 7 7 8 8 8 9 9 9 First ...
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1answer
62 views

Calculating probability of $(n_1,n_2,n_3,\dots, n_k)$-of-a-hands for a generic deck of cards.

Let's say I have a generalized deck of cards, consisting of $R$ ranks, $S$ suits, and with $C$ copies of each card. The number of cards in the full deck would thus be $RSC$, and the number of cards of ...
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1answer
217 views

How to show this conditional expectation?

A factory has produced $n$ robots, each of which is faulty with probability $\phi$. To each robot a test is applied which detects the fault (if present) with probability $\delta$. Let $X$ be the ...
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2answers
41 views

How to sum of this mass density function?

Let $X$ and $Y$ have joint mass function $f(j,k)=\frac {c(j+k)a^{j+k}}{j!k!}$, $j,k\geq 0$ where $a$ is a constant. Find $c$ This sum seems hard to to. How to complete this sum?
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1answer
160 views

Bounding the power of expected value of functions of a random variable.

I am interested in a problem and I do not know where to start looking for possible similar setting. If anyone has a direction to suggest, it would be greatly appreciated. Consider a (finite) set ...
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1answer
461 views

How can I calculate the exact expected value of merge sort comparison (not O(n))?

First, the question stated that I have one unsorted list and then I have to split it out into two lists by fair coin flips. (Ex. Head goes A-list, tail goes B-list) Second, I'm trying to solve the ...
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1answer
65 views

Reflected Brownian Motion

Let $Y(t)$ be a reflected brownian motion. Also let $G(t)$ be a process which keeps count of the number of times $Y(t)$ has hit the value $0$. How do I approach to get distribution of $Y(t)$?
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1answer
1k views

Covariance and variance of a Poisson r.v.

Given a Poisson process $N(t),t\geq 0$ with rate $\lambda$ and another r.v. $T$ independent of $N(t)$ with mean $\mu$ and variance $\sigma^2$, I would like to compute the following quantities: $$ ...
3
votes
1answer
314 views

Number of possible positions for the Rush Hour puzzle

I'm working on a 2-d Puzzle Rush Hour which is a six * six bored that can be filled with various items : 2 blocks length horizontally vertically oriented car - let's call it 1 3 blocks length ...
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2answers
56 views

Probability help!

I am teaching a class of 100 students that has 35 men and 65 women. a.What proportion of the class are men? What proportion of the class are women? Show two different ways to calculate the ...
3
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1answer
276 views

Expected number of different colors

I have a box containing $n$ colored balls. There are $c$ different colors. $k_1$ balls have color $1$, $k_2$ balls have color $2$, ... and $k_c$ balls have color $c$. When I draw $m$ balls with ...
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1answer
68 views

How big is $Z_n^*$?

I would like to find some upper bound on $\frac{n}{|Z_n^*|}$ i.e. to show that many of the elements in $Z_n$ are also in $Z_n^*$. I want to show that $\frac{n}{|Z_n^*|}=O(log^cn)$ for some $c \in N$. ...
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1answer
117 views

How to compute likelihood of drawing specific set of letters from scrabble bag?

Say I have a subset of the standard scrabble tiles $B$ that contains no blank tiles s.t. $|B|=n$ Also consider I have a target set of letters $L$ s.t. $|L|=k$. These are a set of letters not tiles, ...
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0answers
100 views

Finding probability from a markov chain

If I have a markov chain transition matrix for 2 states. Specifically in my case, it is a transition matrix for a bacterial genome with 4 random variables being A,C,G and T. (The bases) If I want to ...
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1answer
46 views

Number of online users

Suppose there are 35 users. The probability that one specific user is online is $\frac{1}{10}$. What is the probability that 11 or more users are simultanious online? I thought that $P(n=\text{number ...
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1answer
63 views

Probability Central Limit Application

Let $X_{1},X_{2},\ldots$ be independent and suppose that $P\left(X_{j}=\sqrt{j}\right)=P\left(X_{j}=-\sqrt{j}\right)=\frac{1}{2}$ for all $j\in\mathbb{N} .$ We want to study the asymptotic ...
3
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1answer
135 views

Asymptotic Distribution by Central Limit Theorem

Let $X_{1},X_{2},\ldots$ be i.i.d. exponential random variables with mean $1$ and variance $1$. Let $$Y_j=\sqrt{j}\left(X_j-1\right)$$ for all $j\in\mathbb{N}$. I want to find the asymptotic ...
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2answers
170 views

Find the supremum for variance

Let $X$ be a random variable such that $\mathbb{P}(X\in[0,10])=1$ $\mathbb{E}(X)=2$ $\mathbb{P}(X<2)\leq1/2$ Find the supremum of all possible values of $\text{var}(X)$. I have some intuition, ...
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1answer
190 views

Weak Convergence to Exponential Random Variable

Assume that $X_1$, $X_2$,... are independent random variables uniformly distributed on $[0,1]$. Let $Y^{(n)}=n\inf\{X_i,1\leq i\leq n\}$. I am asked to prove that it converges weakly to an exponential ...