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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14 views

Maximum value of the intersection of probabilities

I came across a confusing probability problem. It reads as follows: Let $S$ be a sample space and two mutually exclusive events $A$ and $B$ be such that $A \cup B = S$. If $P(\cdot)$ denotes the ...
0
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0answers
16 views

Question about package of cookies that is a random variable

The weight in grams of package of cookies is a random variable with expected value of $300$ grams $\color{blue}{A)}$ assume that X is normally distributed with standard deviation of $15 $ grams ...
2
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0answers
35 views

How long before the prey can escape?

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging ...
1
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0answers
34 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ if I know the density of $(X_2,X_3)$? The ...
0
votes
1answer
42 views

Probability as a function of time

I was really wondering when I have to select any one out of the n options available - the probability of selecting A (let's say) is 1/n. But then I'm confused. When I (or anyone/anything else) bring ...
0
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1answer
19 views

Probability set function of the random variable $X$

Let a point be selected from the sample space $S = (0,10)$. Let $C \subset S$ and let the probability set function be: $$P(C) = \int_C \frac1{10}\ \mathsf dx$$ Define the random variable $X$ by: ...
0
votes
1answer
29 views

Probability of 4-number matching in a lottery in two different situations?

In some lottery, 7 numbers are drawn and each of them from numbers ${\{1, \dots, 45}\}$. To win "Division 6" means to have 4 of 7 drawn numbers. The order of drawn numbers doesn't matter. My ...
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2answers
33 views

Reverse Bernoulli Trial?

I'm struggling to figure out how to do what I think would be called a reverse Bernoulli trial, essentially: How many coin flips must I make to have a 75% change of getting three heads? First of ...
0
votes
1answer
21 views

Probability of an event if the sample space has identical elements

Suppose we have a box, with only one small hole. Suppose 10 distinct black balls and 20 distinct white balls are put in the box. Now, in a random draw of 1 ball, the probability that the ball drawn is ...
0
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0answers
7 views

How to illustrate that the first-fit algorithm for bin packing problem uses STRICTLY MORE bins when only one object becomes larger?

In the bin packing problem, objects of different volumes(lie in $[0, 1]$) must be packed into a number of bins(each of capacity 1). The first-fit algorithm attempts to place the item in the first bin ...
2
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3answers
59 views

Can you get a fair coin flip by rolling a fair, 5-sided die a finite number of times?

Can you get a fair coin flip by rolling a fair, 5-sided die a finite number of times?
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1answer
22 views

Risk Reduction equation

If, after a point in time, your risk of an event falls by 50% in 1 year and then by 100% in 15 years, can someone help me with the equation that will look at your risk reduction to date for any given ...
0
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0answers
13 views

Conditional density of degenerate multivariate normal

Let $X_{12},X_{13},X_{14},X_{23},X_{24},X_{34}$ be identically normal $N(\mu,\sigma^2)$ such that every linear combination among $X_{ij}$'s is also normal, $corr(X_{ij},X_{rs})=\rho$ if ...
1
vote
1answer
38 views

Are infinite-dimensional singletons measurable?

Consider the wiener measure space $C[a,b]$ of all real-valued continuous functions on $[a,b]$ with the wiener measure $\mu$ on the $\sigma$-algebra $\mathcal{A}$ of Carathéodory measurable sets in ...
-1
votes
1answer
51 views

A conditional probability question [on hold]

Let A and B two events and if $P(A)=0.5$ and $P(B)=0.4$ what is the $P(B\mid A^C)$?
3
votes
1answer
25 views

Cancellation law of equal in distribution

I came across this gem while discussing with my friends, If $X$ and $Y$ are two real valued random variables (not necessarily independent) that satisfy $$X =^d X+Y$$ (where $=^d$ means equal in ...
4
votes
2answers
65 views

“Mastermind”-esque safe opening problem.

I read this interview question for a trading job and it seems quite difficult. What is the technique to solving it? You have a safe with six digits and a light. You can input a code, if you have ...
2
votes
2answers
28 views

Statistics question on basil bush random variable

The height, $H$, in meters of a basil bush is a random variable with the probability density function $f_{_H}(t)=e^t,\;0\leq t\leq H_0$ such that $H_0$ is the maximal height. $\color{blue}{(1)}$ I ...
0
votes
1answer
15 views

Inner product estimator - random variable

I'm curently working on the functional space $L^2(\mathbb{R}^n,B(\mathbb{R}^n),\mathbb{P}_X)$ where $\mathbb{P}_X$ is a probability measure. If I generate randomly $N$ realizations of $x_i$ following ...
2
votes
1answer
25 views

Coding Theory - Probability that word received has distance of at most 1?

Suppose the codeword x = 101101 is transmitted over the binary symmetric channel, with symbol error probability p. What is the probability that the word received has distance at most 1 from x? ...
3
votes
0answers
29 views

Convolution of probabilities

It is a well known fact that for a random variable $Z=Y_1+Y_2+...+Y_n$ where $Y_i$ are independently distributed then the probability density function of $Z$ is the convolution of the density ...
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0answers
20 views

Probability of word appearing in a new document given the probability it appears in earlier documents [on hold]

Suppose I have a set of keywords and 10 documents . I can count the frequency and probability of a given keyword occuring in each of these documents . What will be the probability of the same keyword ...
3
votes
1answer
33 views

convergence in probability: speed of convergence

I am not sure if the title appropriately describes the question. I will appreaciate any ideas. Suppose $\{X_n:n\geq 1\}$ is a sequence of random variables defined on a common probability space. ...
0
votes
1answer
20 views

Sums of partially dependent Bernoulli random variables

I am looking for any kind of Chernoff type large deviation bound for the following random variable: $$X = \sum_{i=1}^NX_i$$ where each $X_i$ is an identically distributed Bernoulli random variable ...
1
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0answers
27 views

On the Chernoff bound

Recently, I saw the Chernoff bound written as follows. Let $X_1,X_2,\ldots,X_n$ be drawn i.i.d. on alphabet $\mathcal{X}$ and let $f:\mathcal{X}\to [0,1]$ be any function. Let $\mathbb{E}f(X_1) = ...
2
votes
0answers
20 views

Total variation distance and couplings

The total variation distance between two measures $\mu$ and $\nu$ can be shown to equal the infimum over all couplings $(X,Y)$ where $X\sim\mu, Y\sim\nu$ of $P(X\neq Y).$ What is the supremum of ...
0
votes
0answers
23 views

How many uniform samples are needed to hit every element

Let $D$ be the uniform distribution over $\{1,\ldots,n\}$. How many draws from $D$ (asymptotically) are needed such that with high probability (say $2/3$) all $n$ elements were drawn at least once?
1
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1answer
41 views

Circuit probability question regarding sum of a random number of independent random variables

Suppose we have n circuits that operate in a home. Each one will live according to an exponential random variable with rate λ. If X denotes the time at which a circuit first dies (i.e. the first circuit ...
0
votes
2answers
32 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
0
votes
1answer
28 views

Find probability of a Poisson Process

I have a Poisson process $N(t)$ with $\tau$ for customer arrival in a shop. $N(t)$ is spllitted with two types of arrival (male and female). It can be shown that the process is a combination of two ...
3
votes
0answers
53 views

Why is $F$ continuous?

Why is the function: $F: P(\mathbb R) \to \mathbb R$, $F(X) = \int_X e^{-x} dx$ a continuous function? How to prove such a thing? Does it even make sense to talk about the continuity of such a ...
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votes
2answers
24 views

Conditional day distribution probability

Let $X$ be a random day of the week, coded so that Monday is 1, Tuesday is 2, etc. (so $X$ takes values 1, 2,..., 7, with equal probabilities). Let $Y$ be the next day after $X$ (again represented as ...
2
votes
1answer
34 views

Random variable to the power of minus one?

I have a definition, it goes as follows: $\Pr$ is probability. $X$ is a random variable. $x\in\mathbb{R}$ $$Pr(X = x) = \Pr(\{ \omega\in\Omega \mid X(\omega)=x\})$$ So for example for a dice of 6 ...
1
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1answer
52 views

Coin and Lottery Question

Attempt: For the first part I presume I use bayes' theorem? For the second part, I can't count the number of ways of such a sequence. Thanks.
2
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0answers
18 views

Estimating the Average and Standard Deviation of a Population based on a Sample with Missing Data with Known Ranks

I need a way to shows me how the parameters of PDF, log-normal in this case, can be estimated based on a set with missing data points at the tail end of a sample. For example, Consider we had 20 ...
2
votes
2answers
51 views

Calculating probability for forming a triangle

I am having trouble coming up with a solution for this problem: There is a stick of unit length. We break it into two parts. Now, we pick the bigger one and break it into two parts. I want to ...
2
votes
3answers
38 views

Distribution of a fractional part of the sum of uniform RVs

I had a question in class not long ago which I couldn't solve. I've been digging into it for a few hours now but I can't find the right direction. So the question is: Let $ U_1,..,U_n$ be I.I.D ...
0
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0answers
14 views

Simulate ICA Source Signal

I am using the fastICA package in R for a matrix of time series information. However, if I wanted to simulate the process for risk management purposes how exactly could I do this? For example lets ...
0
votes
2answers
27 views

The random variable $ Z = 1-F(X)$

I will formulate the theorem (with no proof) if $X \in \mathbb{R}$ is a random variable with continuous distribution function $F$ then the random variable $Z = 1-F(X)$ has a uniform distribution on ...
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votes
2answers
30 views

Does the order in a circular arrangement matter?

I posted a question a while ago: Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs. My question here is: imagine a ...
3
votes
1answer
61 views

Play until one is bust

If player A and B have $a$ and $b$ millions pounds respectively, where $a,b$ are natural numbers. They play a series of games in which the winner receives one million pounds from the loser (draws ...
1
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1answer
32 views

Custom Weighted Formula

I'm in need of a mathematical formula that will be ultimately utilized in any programming language that would give me a value that I could ultimately sort or rank by. I have 2 variables: variable 1 = ...
2
votes
2answers
377 views

Stars and Bars vs PIE

I randomly made up this question so I could check: There are $3$ kids and $6$ gifts, how many ways to distribute so that each kid has at least one gift. Obviously, $**|**|**$ there are ...
3
votes
1answer
29 views

Poisson Process Derivation.

I was looking at a derivation for the poisson process , which tells the number of events occurring in time $t$ , I came across the following differential equation : $\frac{d}{dt}(P_n(t))$ = ...
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votes
0answers
15 views

Combining experts probability judgements [on hold]

I seek for function that combine list of probabilities of event into one probability value. Each probability is given by expert. This function should fulfill: $Pc(\{1.0\}) < Pc(\{1.0,1.0\})$ ...
0
votes
1answer
43 views

Find the CDF of a function of two random variables

The joint probability density function of two continuous random variables $X$ and $Y$ is: $$f(x,y) = \begin{cases} 6x,& 0\leqslant x\leqslant y,\ 0\leqslant y\leqslant 1\\ 0,& \text{ ...
2
votes
1answer
25 views

Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
1
vote
0answers
12 views

conditional density function with respect to a statistic.

I have read a formula in a textbook: $$f_{Y|S}(y|s;\theta)=\frac{f_Y(y;\theta)}{f_S(s;\theta)}$$ $Y$ is a random vector and $S$ is a statistic of $Y$. According to the definition of conditional ...
1
vote
1answer
33 views

Definition of n independent event and example

Given a finite set of events $A_1,\dots,A_n$, the events are said to be independent if and only if for any subset of indices $I$ we have: $$\mathrm{P}\left(\bigcap_{i\in I} A_i\right)=\prod_{i\in I} ...
9
votes
3answers
1k views

Kelly criterion with more than two outcomes

I want to calculate the Kelly bet for an event with more than two possible outcomes. Suppose the following game: A jar contains $10$ jelly beans. There are $7$ black jelly beans, $2$ blue jelly ...