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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6 views

Statistical distance between a multiplicative mask and a random number

Given $x \in \{1,\ldots,2^n\}$ and a uniform random $r \in \{1,\ldots,2^{n+k}\}$, then the statistical distance $\Delta(x + r\bmod q; r) < 2^{-k}$, for a $q > 2^{n+k+1}$. With addition this is ...
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0answers
12 views

Functional Equation of Probability Distributions

Suppose you have a real random variable $X$ that has probability distribution $f_X$ meaning $$ P(\alpha \le X \le \beta) = \int_{\alpha}^{\beta} f_X(x) \ dx $$ Now assume $\Phi(f_X)$ is also a ...
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2answers
22 views

Variance of the random sum of a Poisson?

We have that $N$ and ${X_1, X_2, \dots }$ are all independent. We also have $E [X_j] = \mu$ and $Var[X_j] = σ^2$. Then, we introduce an integer–valued random variable, N, which is the random sum such ...
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0answers
16 views

Multiple lottery probability

Say there are 5 lotteries (A,B,C,D,E) with a 100 prize. That you have 5 dollars All tickets cost one dollar each That you want to buy 5 tickets. One thousand tickets in total are sold in each ...
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0answers
29 views

The probability of the sum of $10$ dice rolls adding up to $57$

So the question is: given that you roll $10$ dice, what is the probability of the sum of the total dice rolls adding up to $57$? I know that there are three ways to do this: Seven die rolls must ...
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2answers
12 views

Find the moments of a binomial conditioned on a binomial?

Suppose that $Y$ has the binomial distribution, $Bin(20, 0.25)$ and conditioned on $Y$, a random variable $X$ that has the binomial distribution, $Bin(Y, 0.5)$. How can I derive the $k$th moment of ...
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2answers
15 views

Find the joint and conditional distributions of $Z=X+Y$?

Suppose that $X$ and $Y$ are independent and identically distributed: $$P (X = k) = P (Y = k) = ρ (1 − ρ)^k$$ for $k = 0, 1, \dots$ and let $Z := X + Y$. Find the joint distribution of $(X, Z)$ ...
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3answers
32 views

Meaning of $P(X \in A)$

I have the following problem. I'm struggling a little bit with the expression $P(X \in A)$. My problem is that $A$ is a set, whereas $X$ is a function. I can not really related this two items. Here ...
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0answers
21 views

Explicit formula for return probability of simple random walk

Is there an explicit formula for the probability that a simple symmetric random walk on $\mathbb{Z}$ starting from $1$ will not hit $0$ before time $t$?
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0answers
15 views

$r$ balls are randomly distributed into $n$ urns. What's the expected number of urns with $k$ balls?

My text book uses the linearity of the expected value to compute it. It defines a random variable $X_i$ that indicates whether the urn $i$ contains $k$ balls or not. So the asked value is $E[X_1 + X_2 ...
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1answer
29 views

Exploding dice in a dice pool

Say we role $n$ identical, fair dice with $d$ sides (every side comes up with the same probability $\frac{1}{d}$). On each die, the sides numbered from $1$ to $d$ with no repeating numbers on any one ...
2
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0answers
16 views

Return time for two independent one dimensional random walks

Let $X^1$ and $X^{-1}$ be two simple random walk in $\mathbb{Z}$ starting respectively from $1$ and $-1$. Let $\tau$ be the first time one of them reaches the origin, $$\tau = \inf \{ j \geq 0 \, : ...
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3answers
29 views

Help with finding the probability of this exam question

I need help with solving one of the questions the teacher gave us to prepare for an upcoming exam. I tried solving it but with no luck. Here is the question: On one shelf there are 5 hardcover ...
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0answers
19 views

Discrete random vector and their sum

Given the random vector $(X,Y)$ with joint probability $P(0,1)=\frac{1}{18}$, $P(1,2)=\frac{3}{18}$, $P(1,4)=\frac{5}{18}$, $P(2,0)=\frac{2}{18}$, $P(2,1)=\frac{4}{18}$, $P(2,3)=\frac{3}{18}$ and ...
1
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1answer
17 views

There are 6 white balls and 9 black balls. Probability of drawing two white, then two black?

From A First Course in Probability (9th Edition): 3.5 An urn contains 6 white and 9 black balls. If 4 balls are to be randomly selected without replacement, what is the probability that the ...
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0answers
10 views

mutual information and data processing inequality for $X\to Y\to Z$ where $Y=f(X)$

Let $X\to Y\to Z$ be three random variables. The data processing inequality states $I(X;Y)\geq I(X;Z)$. Further assume $Y=f(X)$ where $f:\mathcal{X}\to\mathcal{Y}$ is an arbitrary function. What ...
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votes
1answer
14 views

Inclusion-exclusion error clarification

Suppose you pick a number between $1$ and $30$ uniformly at random. Let $A$ be the event that the number is even. Let $B$ be the event that the number is divisible by $3$. Let $C$ be the event that ...
0
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1answer
23 views

Total probability law clarification

Suppose you roll a fair 6-sided dice three times. There are $6^3$ possible outcomes and each is equally likely. Let $A_1$, $A_2$, $A_3$, $A_4$, $A_5$, and $A_6$ be the events that the last value is a ...
0
votes
1answer
22 views

Find the probability that two samples contain all different balls.

Suppose we have a box containing $n$ balls numbered $1, 2,\dotsc,n$. A random sample of size $k$ is drawn without replacement and the numbers on the balls noted. These balls are than returned to ...
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0answers
16 views

Preserving independence of random variables

Suppose I have three random variables, $X,Y,Z$ with $X$ independent of $Z$, $Y$ independent of $Z$. Which transformation can I apply to $X,Y$ to that the result is again a random variable independent ...
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1answer
25 views

There are $n$ seats in a room. If $n$ people come to the room, what is the probability that $j$ specified people occupy $j$ specified seats?

There are $n$ seats in a room. If $n$ people come to the room, what is the probability that $j$ specified people occupy $j$ specified seats? ($j$ names were tagged on the $j$ seats) $n$ people can ...
2
votes
1answer
35 views

If X and Z are independent and Y and Z are independent random variables, is cov(XY, Z) = 0?

Let $X$, $Y,$ and $Z$ be random variables. (There are no restrictions on these variables, but you may assume that these are continuous random variables if you want.) Suppose that $X$ and $Z$ are ...
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1answer
22 views

Covariance of dice tosses that result in 1 or 2 (fake proof)

Question: Consider n independent tosses of a $k$-sided fair dice. Let $X_i$ be the number of tosses that result in $i$. What is the covariance $\mathrm{cov}(X_1,X_2)$ of $X_1$ and $X_2$. ...
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2answers
29 views

Three People Rolling a fair Die.

Three players A, B and C take turns to roll a fair die; they do this in the order ABCABC... (a) Find the probability that, of the three players, A is the first to throw a 6, B is the second, and C is ...
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0answers
24 views

Expected Value Given pdf [on hold]

Suppose that fifteen observations are chosen at random from the pdf $ f_Y(y)=3y^2$, 0≤ y ≤1. Let X denote the number that lie in the interval [1/2 , 1]. Find E(X).
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0answers
10 views

Matching of points in two discrete linear sequences with potentially missing points

This is a question that I've been thinking about in my research lately. I've gone down the route of a few linear-optimization techniques, but nothing particularly spectacular has come up. Anyway, ...
4
votes
2answers
71 views

Can someone explain what a portfolio is in financial math?

I took mathematical probability last semester and now I am taking financial mathematics, but only probability was a pre requisite for financial math (no finance classes were required). These types of ...
2
votes
1answer
10 views

Moment generating function of sample mean of bernoulli random variables

Let $p \in (0,1)$ and $n \in \mathbb{N}$. We consider a sample of $n$ i.i.d. Bernoulli variables $X_1,\dots,X_n$ with parameter p. Computer $E[e^{\lambda\bar{X_n}}]$ such that $\bar{X_n}= \frac{1}{n} ...
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0answers
10 views

How to do the inverse problem of the kernel density estimation

Given $x_1, x_2,..x_n ; x_i \in R$ that drawn from an unknown distribution $P(x)$ and a constant $ C$ $ 0 \leq C \leq 1$. Find $x^{*}$ such that $$P(x^{*}) =C$$. We want to use the kernel density ...
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6answers
49 views

Probability of getting $5$ heads on $10$ (fair) coin flips?

Even before attempting the problem, I immediately defaulted to an answer: $\frac{1}{2}$. I thought that this was a possible answer since the probability of flipping a head on one flip is definitely ...
2
votes
2answers
24 views

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable?

If I have that $\limsup_{n}E|X_n|^{r} \leq E|X|^{r}$, is that enough to show that $\{|X_n|^{r}:n\geq 1\}$ is uniformly integrable? I am not sure here if the limsup condition here is as strong as if I ...
4
votes
2answers
35 views

Probability of island having 8 people born with disease, estimate?

The chances of being born with a certain disease are estimated as $1$ in $1200$. What is a good estimate of the chance that an island with $10000$ inhabitants has precisely $8$ people born with that ...
4
votes
2answers
32 views

What is the probability that these two objects are of the same color?

We have $11$ bins with $10$ objects each. Every object is either black or white, and the $i$th bin ($1 \le i \le 11$) has precisely $(i -1)$ black objects in it. Someone selects, uniformly at random, ...
0
votes
1answer
15 views

Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill. [on hold]

Three people have been exposed to a certain illness. Once exposed, a person has a 50-50 chance of actually becoming ill. a) What is the probability that exactly one of the people becomes ill? I am a ...
2
votes
1answer
31 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
4
votes
1answer
22 views

$p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent, then $p(X \wedge Y \wedge Z) > 0$?

Is the following statement true or not? Let $X$, $Y$, $Z$ be $3$ events in the same sample space such that $p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent. Then $p(X ...
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0answers
10 views

probability that a customer who purchases up to $5$ songs from $4$ music genres prefers jazz and buys at least $3$ songs [on hold]

Customers can choose from $4$ music genres: jazz, rock, new age, country; and can purchase up to $5$ songs. The events are: $A =$ customer prefers jazz and buys at least $3$ songs $B =$ the customer ...
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votes
1answer
19 views

Conditional Independent clarification

Let's say I have $3$ events with probabilities $P(A) = 0.5, P(B) = 0.5$ and $P(C)= 0.5,$ and I need to find if $$P(A \cap B \mid C) = P(A \mid C)P(B \mid C)$$ I am tying to prove this by expanding ...
0
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0answers
23 views

integral and probability

Let $N_t$ be a Poisson process and $S_{N_t}=X_1+...+X_{N_t}$. Let $A_t=t-S_{N_t}$ and $B_t=S_{N_t}-t$ 1) Show $P(B_t \geq x \ \text{and}\ A_t \geq y)=\frac{1}{E(X_1)} \int_{x+y}^{\infty} P(X_1 \geq ...
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votes
2answers
20 views

Conditional probability with dependent events

We have 2 dice. One is fair. The other one lands by the following probabilities: 6: 1/2 5: 1/10 4: 1/10 3: 1/10 2: 1/10 1: 1/10 We roll both dice. What is the ...
1
vote
1answer
19 views

Conditional Probability clarification

Here's a sample problem: Before each workout, I either drink a cup of coffee, a gatorade, or a cup of water. The probability of coffee is P(C) = 0.6, the probability of gator is P(G) = 0.3, the ...
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0answers
34 views

An urn contains $nr$ balls numbered $1,2..,n$

An urn contains $nr$ balls numbered $1,2..,n$ in such a way that $r$ balls bear the same number $i$ for each $i=1,2,...n$. N balls are drawn at random without replacement. Find the probability that ...
2
votes
1answer
20 views

Win/Lose ratios and selection strategies

Imagine the following scenario: You're on a TCG tournament which allowed you to bring N decks with you. After each game, you might select another deck for your next game. You are allowed to keep ...
1
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2answers
44 views

Exponential distributions [on hold]

Good evening to all, I'm so much confused about a question; Assume there is a workshop with two machines. The times until the failures of machines $1$ and $2$ are independent and exponentially ...
0
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1answer
22 views

Find the probability of B

Suppose you roll a fair 6-sided dice three times. There are $6^3$ possible outcomes and each is equally likely. Let $A_1, A_2, A_3, A_4, A_5,$ and $A_6$ be the events that the last value is ...
0
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1answer
34 views

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads

A coin is tossed $m+n$ times. Find the probability of getting atleast $m$ consecutive heads I already know that the exact same question has already been answered here But I am trying to solve it ...
2
votes
2answers
25 views

chain rule conditional entropy

I have to prove the chain-rule for conditional entropy. I kept getting stuck on one step, so I looked up a proof and found this: \begin{align}H(Y\mid X)&= \sum_{x\in\mathcal X, y\in\mathcal ...
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2answers
12 views

Possible orderings when the items are not unique?

First of all, I'm sure this question has been answered somewhere on the web, but I am just starting probability and I don't have the vocabulary to know what to look for, which is why I am asking here. ...
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0answers
7 views

Is the result of a Monte-Carlo simulation of a continuous function and with continuous input distributions again continuous?

Is the result of a Monte-Carlo simulation of a continuos function and with continuos input distributions again continuous? Suppose, we have a continuos function $f$ and a number of continuous random ...
0
votes
1answer
28 views

Equality in Conditional Jensen's Inequality

Conditonal Jensen's Inequality says that for a convex function $\varphi$, a random variable $X$, and a sub-sigma-field $\mathcal{F}$, $E[\varphi(X)\mid \mathcal{F}] \geq \varphi(E[X\mid ...