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
42 views

Sets and probability

The number of total outcomes of an experiment are $25$. If $A$ and $B$ are two non-empty independent events of the experiment such that outcomes in favour of event $A$ are $15$, then the minimum ...
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2answers
52 views

Profitable strategy in coin tossing?

There is a coin with a probability $p$ of heads, and $1-p$ of tails. Tosses are independent of each other. When you bet an amount of money $x$, you receive $2x$ if it lands heads, and you lose what ...
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1answer
18 views

Chance of drawing 4 red marbles out of a big bag.

In a bag with an infinite number of marbles, where a third are red, a third are green and a third are blue. Given that you pick $10$ marbles, of which $3$ are blue, what are the chances of picking $4$ ...
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2answers
30 views

What is the probability of observing three or fewer 6s when rolling a fair die twenty times? [on hold]

What is the probability of observing three or fewer 6s when rolling a fair die twenty times? I am trying to figure this out..please be detailed on explaining. Thank you!
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0answers
22 views

Conditional probability with sigma-field [on hold]

If $\mathcal{F}_1, \mathcal{F}_2$ and $\mathcal{G}$ are $\sigma$-field, we say $\mathcal{F}_1$ and $ \mathcal{F}_2$ are conditionally independent give $\mathcal{G}$ if $$ P(A\cap B\mid ...
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1answer
27 views

Why total probability is the sum of conditional probabilities?

Consider the following question taken from this link, question number $25$: We have four boxes. Box $1$ contains $2000$ components of which $5$ percent are defective. Box $2$ contains $500$ ...
0
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1answer
21 views

If I define a series of random variables to be $X_n \sim N(0,n)$, does $X_n$ converge almost surely to any limits?

If I define a series of random variables to be $X_n \sim N(0,n)$, does $X_n$ converge almost surely to any limits? Intuitively this is not the case, but I am not sure how to formally show this, would ...
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2answers
17 views

Estimate on Probability of a standard normal variable

In the book written by Karatzas & Shreve, at the page - 111; the authors have mentioned about a result: If $Z_{v}$ be a standard normal variable; then for $\epsilon \gt 0$ ; $\mathbb P ...
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0answers
21 views

What is an example where we have convergence in distribution to a constant, but that doesn't imply convergence almost surely?

I have been trying to disprove that if I have a sequence of random variables $X_n$, that $X_n \to a$, where $a$ is a constant, in distribution doesn't imply $X_n \to a$ almost surely. One example I ...
0
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0answers
27 views

Ito Formula for Stochastic Integral

Suppose I have $$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$ What would be the process satisfying the following process of $y_t$? $$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$ I'm not quite sure ...
2
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1answer
29 views

Production line probability problem

A manufacturer has six distinct motors in stock, two of which came from a particular supplier. The motors must be divided among two production lines, with three motors going to each line. If ...
2
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1answer
70 views

Find $\int\limits^{\infty}_{0}\int\limits^{\infty}_{0}{\frac{1}{(x+y)^{3/2}}\exp\left\{-\frac{a^2}{2(x+y)}\right\}}\,dy\,dx$.

In my posterior probability computation, I got the following integration and I could not figure it out. ...
0
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2answers
46 views

Find probabilty

I have this table of information: Probabilities: \begin{array}{c|c} .919 & ????\\\hline ???? & .274 \end{array} How do I find the probabilities of the question marks? I thought each row ...
0
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2answers
27 views

Is it true in general that $\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r$?

If I have that $X$ is a random variable, for $\epsilon > 0$, and $r \geq 1$, is it true that: $$\int_{|X| \leq \epsilon} |X|^r \, d\mathbb{P} \leq \epsilon^r.$$? If so, is there a reason why? ...
-1
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0answers
23 views

Probability of two people 10 minutes walk away in the same village working together in a different country. [on hold]

I am a teacher of English about to start a new job at a language school in Wroclaw, Poland. My native home is situated in a small village called Kemsing in South East England. Shortly after meeting ...
0
votes
1answer
37 views

Two candidates, A & B, are running for president. What is the probability that candidate A beats candidate B?

Candidate A has already garnered 80 votes. Candidate B has already garnered 50 votes. The number of votes a candidate must have in order to win the election is 115. The votes of 5 states are still ...
1
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3answers
30 views

Joint probability uniform distribution

I have a question on finding probabilities of joint distributions, specifically two independent random variables that are Uniformly distributed. The question I wish to solve is this one: We agree ...
0
votes
1answer
16 views

Show that Uniform$(1,5)$ is neither singular nor absolutely continuous with respect to Uniform$(0,3)$.

Actually, I'm just studying singular continuity, absolute continuity.I know the definitions.And have solved few very basic sums. Now, in this problem, I'm not understanding what does this 'with ...
2
votes
3answers
44 views

Largest possible value of $P(A \cap B)$

Suppose $A$ and $B$ are events with $P(A)+P(B)>1$. Show that the largest possible value of $P(A \cap B)$ is $ \min(P(A), P(B))$. I suspect I'm supposed to use $P(A \cap B) = P(A)+P(B) -P(A ...
1
vote
2answers
39 views

Calculating first and second moments for random sums?

Assume that $N$ and $X_1, X_2, \ldots $ are all independent and identically distributed over $(0,1)$ with the density function: $f (x) = cx^2 (1 − x)^2$. An integer–valued random variable, $N$ ...
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3answers
37 views

Find $f_Y(y)$ if $f_{X,Y}(x, y)=2e^{−x}e^{−y}$ for $x$ and $y$ defined over the shaded region.

I have no idea how to start this problem. Any help would be greatly appreciated. Thanks
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0answers
12 views

Reduction of Two Independent Random Variables in Quadratic Form

Consider the $n \times 1$ random vector $\mathbf{x}$ and the $p \times 1$ random vector $\mathbf{y}$. The vectors are independent of each other, and $\mathbf{y}$ has an expected value of zero. I want ...
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0answers
8 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
24 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 ...
2
votes
3answers
83 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 $\operatorname{E} [X_j] = \mu$ and $\operatorname{Var}[X_j] = σ^2$. Then, we introduce an integer–valued random variable, $N$, ...
0
votes
1answer
58 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 ...
2
votes
2answers
33 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
26 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
33 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
27 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
20 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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votes
3answers
205 views
+50

Exploding dice in a dice pool

Say we role $n$ identical, fair dice, each with $d$ sides (every side comes up with the same probability $\frac{1}{d}$). On each die, the sides are numbered from $1$ to $d$ with no repeating number, ...
2
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0answers
23 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
40 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
22 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
24 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
28 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
19 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
votes
1answer
32 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
61 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
21 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 ...
0
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1answer
47 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
41 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 ...
0
votes
1answer
26 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
39 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
53 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), where E(X) is the expected ...
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2answers
83 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
14 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} ...
0
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0answers
12 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 ...
1
vote
6answers
54 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 ...