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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2answers
1k views

Derivation of the density function of product of two random variables

I am looking for distribution of product of two random variables. Best I could found so far was this formula from the relevant Wikipedia page: $$ f_Z(z) = \int_{-\infty}^{+\infty} \frac{1}{|x|} ...
2
votes
2answers
78 views

$\mathbb{E}[X^+]$ and $\mathbb{E}[X^-]$

I was wondering what is the relationship between $\mathbb{E}[X^+]$ and $\mathbb{E}[X^-]$, when $\mathbb{E}$ is a sublineair expectation and $\mathbb{E}[X] = \mathbb{E}[-X] = 0, \mathbb{E}[X^2] > - ...
4
votes
0answers
87 views

Does Multiplicative Version of Azuma's Inequality Hold?

We know that there are multiplicative version concentration inequalities for sums of independent random variables. For example, the following multiplicative version Chernoff bound. Chernoff bound: ...
2
votes
2answers
318 views

Statistics Probability Help

I just began to take this stats course in HS and I'm a little stuck on these 2 problems below. Can anybody please help me out with the solutions? Thank you. Anything is appreciated. Let $Y$ be a ...
3
votes
1answer
531 views

What is the intuition behind the Borel Cantelli Lemma?

The Borel Cantelli Lemma states that the probability of a set of outcomes happening infinitely many times in an infinite sequence is $0$ under some conditions. I neither understand the math (behind ...
2
votes
7answers
583 views

Probability of an event after time has passed

A friend of mine posed this problem and we have had a disagreement on the answer. The problem: There is a 90% chance that some event will happen in the next year. There is a 95% chance that the ...
1
vote
2answers
39 views

Probability of number of events where $p_{n+1} = \frac{1}{5}p_n$

For $ n \ge 0, p_{n+1} = \frac{1}{5}p_n $. Where $p_n$ is the probability of an envent occuring n times in a period. What is the probability that more than one event occurs?
2
votes
2answers
105 views

gambling probability problem

We are given a fair coin. We start out with 5 dollars. We keep tossing the coin. If the outcome is different than the previous one, we are awarded another 5 dollars. However, we do not get anything if ...
1
vote
1answer
272 views

Maximum Likelihood fitting of truncated, mixed, two population systems (Gaussian Examples)

TLDR: I am trying to do maximum likelihood fitting of a dataset having two mixed populations, observed over a subset of their parameter space, within it to two pdfs. I include working code with ...
2
votes
2answers
161 views

probability: finding Probability density function

Let $X\sim\exp(λ)$ and $Y$ equals its decimal part. How would you find the probability density function of $Y$? I started by looking for $F_Y(y)$ but got stuch in this level: $F_Y(y)=P(Y\le ...
1
vote
1answer
111 views

Probability of an exact number events over several periods

The probability of an event occuring N times in a day is $ P[N=n] = \frac{1}{2^(n)} where 0\le n $. The number of times an event ocurrs in one day is independent of the number of times the event has ...
2
votes
1answer
75 views

Equality of two definitions of conditional expectation

Let $X,Y$ be two random variables and let $Q(x,B)$ be a transition kernel from $X$ to $Y$: $$ \mathsf P_{X,Y}(A,B) = \int\limits_{A}Q(x,B) \, \mathsf P_{X}(dx) $$ Then we can define $\mathsf E(Y ...
1
vote
2answers
135 views

Average error of two normally distributed measurements

There are two methods of measuring on object of length $x$. The error of the first method is normally distributed with a mean of 0 and standard deviation of $0.0056x$. The error made by the second ...
2
votes
1answer
443 views

Computing an integral involving standard normal pdf and cdf - with peculiar limits.

I have had a look at some of the other questions on this topic but cannot quite work out the solution to this integral (or prove that there isn't a solution). Is there a way to work out: ...
0
votes
1answer
69 views

Optimal $p$ for biased coin?

You are given a biased coin with probability $p$ of getting $H$ and $1-p$ of getting tail. Each flip is independent of another. We keep flipping the coin until we get $4$ consecutive tails. For each ...
3
votes
2answers
148 views

How to differentiate discrete probabilities?

Assuming that I have a function $$f(p(x),p(c),p(x,c)) = \ln (p(x)p(c)) + \ln (p(x,c))$$ where $p(\cdot)$ are discrete probabilities, $x \in X, c \in C$ are random variables. So $p(x)=p(X=x)$ denotes ...
1
vote
0answers
119 views

What is the probability of a hash containing a substring?

An SHA-$ 256 $ hash is one that is $ 64 $ digits long, where every digit is a hex value ($ 0 $-$ 9 $ and a-f). Here’s an example hash: ...
13
votes
1answer
353 views

What is the probability of having a pentagon in 6 points

If the probability that $5$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$ occur to be the vertices of a ...
2
votes
0answers
762 views

Probability of tossing a biased coin without having k heads consecutively in a row

I got asked by a friend this question; I have a coin, the probability of receiving a head by tossing is $p$ and tail $1-p$. I have to toss it $n$ times without getting $k$ heads in a row. What is the ...
6
votes
1answer
4k views

Expected number of calls for bingo win

Before I begin, I did a search through math.stackexchange and came across two previous attempts to get people to solve probability problems involving bingo. Neither produced a response. So what ...
1
vote
1answer
134 views

Probability that two people share the same Amino Acid sequence given these parameters.

This question will sound oddly specific like it's my homework but it's not. It's my research and I could use some help thinking about it. A person has 20 amino acids to choose from in constructing a ...
1
vote
1answer
113 views

Computing conditional probability combining events

I know $P(E|A)$ and $P(E|B)$, how do I know $P(E|A,B)$? Assuming $A,B$ independent.
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votes
0answers
92 views

Modeling concurrent internet users

I'm feeling generous in the new year and want to open my Wifi connection to the public. I want to estimate the effect that $N$ additional users on my router would have on download times. In other ...
1
vote
1answer
369 views

Probability of failure

An electronic product contains $8$ circuits. The probability that any of the circuits is defective is $0.10$, and they are independent. The product operates only if at least two of the circuits are ...
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2answers
94 views

Probability problem found in textbook

"Shown is the board for a simple dice game. You roll a die and move the same number of squares (for example if your first roll is a $3$, move to the $3$ square). If you land on an arrow's tail, you ...
5
votes
2answers
1k views

Poisson distribution with exponential parameter

I don't know how to solve Exercise 8, Section 5.2 from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001. For those who don't have this book: ...
1
vote
1answer
166 views

An interesting probability calculation problem

This is my first question in this site. I'll be glad if you respond to this question. I am working with an interesting probability calculation problem. In my problem there are ratings for 32 chess ...
1
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3answers
244 views

Question on overlapping sets

Let's say that four of us (persons A, B, C, and I) are each handed our own, complete deck of cards. All four decks are identical, meaning that an Ace of Hearts in B's deck is equivalent to (and ...
0
votes
1answer
259 views

What is the formula to predict a high probability of range 0~9, given by past results?

I'm very new and i'm below average for my math. But there's this thing about a lucky draw game that bugs me to think of a probability. Say the host of a party has already drawn several luck draw ...
2
votes
1answer
1k views

If $X$ is uniform on the interval $[0, 2]$ , find the PDF of $X^2-2X$

If $X$ is uniform on the interval $[0, 2]$ find the PDF of $Y = X^2-2X$. I solved for $X$ and got $x=1 + \sqrt{1+y}$ or $x=1 - \sqrt{1+y}$. Not sure what to do from here.
6
votes
1answer
219 views

A Friend of a Friend: Transitivity in Large Random Graphs

Suppose we have a network $G$ of $n$ people, who share $M$ connections. Define the transitivity index $\tau(G)$ as the count of all possible threesomes having $3$ connections divided by the count of ...
0
votes
1answer
87 views

Determining probability in a random loop

[Note: I hope the title is appropriate. I am very much mathematically ignorant except when it comes to a limited scope of computer programming. Also, please excuse my lack of proper notation, and ...
0
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1answer
178 views

Give examples of not independent random variables which are uniform s.t. $P(X+Y=1)=1$ and $X+Y$ which is uniform in the interval $[0,2]$

Give examples of (not independent) random variables $X$ and $Y$, both of which are uniform in the interval $[0, 1]$ and such that $\mathbb{P}(X + Y = 1) = 1$ $X + Y$ is uniform in the interval $[0, ...
2
votes
1answer
68 views

Linear system with probabilities (algebra)

I have a small problem that delays my project , it seems I am stuck here loads of time. It is probably very easy but I cant see it right now , I am very anxious about this, please take a look: ...
0
votes
1answer
53 views

Bayes theorem - probability

A system sends 1s and 0s in sequence with equal probability. Errors are made, so the probability of receiving a digit $r_j$ (equal to 0 or 1) given that a certain digit $s_i$ (equal to a 0 or 1) was ...
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vote
2answers
228 views

Where is the logic with this trinomial probability?

I am studying for Exam P to take in a few hours so I'd appreciate very much if someone got to me befoe I left. The problem says that we have three categories of risk, high, medium and low risk. The ...
15
votes
1answer
299 views

Expected size of subset forming convex polygon.

If there are $4$ random points in the plane whose horizontal coordinate and vertical coordinate are uniformly distributed on the interval $\left(0,1\right)$, what is the expected largest size (or ...
0
votes
1answer
1k views

Probability density of lightbulbs (Independent events)

A lamp has two bulbs, each of a type with an average lifetime of 5 hours. The probability density function for the lifetime of a bulb is $\sigma(t)$, What is the probability that both of the bulbs ...
0
votes
1answer
33 views

Does the difference of the total happened count of two incidents of a bernoulli distributions also follows certain distribution?

Oh, it is hard to express math questions in English. What I mean concerns a Bernoulli distribution $B(n,p)$. We know that $A$ happens $k$ times and $A^c$ happens $n-k$ times. So does the difference ...
4
votes
1answer
96 views

Monthly rental fee to achieve given profit on average, given probabilities of numbers of rentals

I have this problem here and I'm very unsure of how to start this. I have an idea but I'm not sure where to go from a certain point. The problem says: A video rental store is analyzing a flat fee ...
2
votes
2answers
89 views

Geometric Distribution question

A motor insurance company has sold $150$ insurance policies. Let $N_{i}$ represent the number of claims made on policy $i$. You may assume that $N_1, N_2,..., N_{150}$ is a sequence of independent ...
0
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0answers
438 views

Reliability and probability

A system comprises two subsystems operating in series. The first subsystem has two components that operate in parallel. The reliabilities of the components in both subsystem is $r$. Assume that the ...
1
vote
3answers
72 views

Geometric distribution, tossing a die [duplicate]

Possible Duplicate: geometric distribution throwing a die Yesterday I posted a question which was answered but I disagree with the answer so I'd like to ask again so we can discuss it ...
7
votes
3answers
219 views

Stein's lemma condition

(Apologies if I break some conventions, this is my first time posting!) I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable ...
0
votes
0answers
333 views

Expectation of a joint distribution or RVs X Y Z

How does E[XYZ] breakdown for correlated discrete RVs X Y and Z (or more than two, at least)? Is there a link to a full example somewhere? What I see is that E[XYZ] = SumX[SumY[SumZ [ (XYZ) (jpmf of ...
0
votes
1answer
197 views

Finding the standard error of mean

Suppose you have a process $x_{t} = \mu +w_{t} -0.8*w_{t-1}$ where $w_{t}$ ~ $wn(0,\sigma_{w}^2)$. How do I calculate the standard error of $var(\bar{x})$ for estimating the mean. I know: ...
1
vote
1answer
180 views

autocovariance function for nonlinear time series

I want to find the autocovariance function of $y_{t} = \exp(x_{t})$ which is a nonlinear time series. I am assuming $x_{t}$ is stationary with mean $\mu$ and covariance $\gamma(h)$. When I try to ...
0
votes
0answers
67 views

Show the following random variables in $\mathbb{R}^2$ have the same distritbution

$X_1, X_2, \cdots, X_n$ are independent Gaussian $N(0,1)$ random variables. I need to show that the following random variables in $\mathbb{R}^2$ have the same distribution: $\displaystyle ...
2
votes
1answer
62 views

simplify expectation definition Hidden Markov Model

I am reading Rabiner's paper entitled "A tutorial on hidden markov models and selected applications in speech recognition". There is a very simple example where he simplifies the calculation of an ...
0
votes
2answers
6k views

what is the mean of probability density function

Suppose we have a PDF $$p(x)=\frac{1}{2}\left[\frac{1}{\sqrt{2\pi}}\left(\exp \left(\frac {-(x-1)^2}{2}\right) + \exp \left(\frac {-(x+1)^2}{2}\right)\right)\right] \quad\text{for}\; -\infty < x ...