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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0answers
8 views

How does conditional expectation really operate?

Let there be a keyboard with k keys, only 9 of which are numbers, which are 1,...,9. A monkey performs a series of random taps. The series will end as the monkey taps a non-number key. Let $N$ be the ...
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2answers
19 views

Question about probability distributions

I've recently came across this question: ...
0
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0answers
17 views

What is the probability of an event happening in some interval given probability of it in x interval?

Suppose there is an event that happens with a probability of y in x interval of time, what would be the probability of it happening in x/2 interval of time? Would that be y/2 or is there something ...
-1
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0answers
21 views

Can someone confirm if the solutions are correct?

For the first one, I did 163 / 1200.. For the second one, I got 24% I think both of mine are correct, but solutions say otherwise.
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5answers
38 views

Picking (and replacing) among five balls in an urn

An urn contains 5 balls numbered from 1 to 5. A ball is chosen at random and its number is noted the ball is then returned to the urn. this is done a total of 5 times. What is the probability that ...
0
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0answers
18 views

How do I calculate conditional PDF?

Obtain $$P(2 < Y < 3 | X = 1)$$ where the joint pdf of X and Y is $$f_{X,Y}(x,y) = (6-x-y)/8$$ where $$0 < x < 2$$ and $$2 < y < 4$$? so first, I did $$f_Y|X=1(y) = ...
1
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3answers
56 views

Probability that two numbers differ by one bit

Assuming that t is the bit length of the numbers and that we can pick 2 random numbers (the same number cannot be chosen twice), which is the probability that the two numbers will differ by exactly ...
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1answer
30 views

overlapping two events in one year with e certain duration. [on hold]

I'm struggling with the following problem: Given that two events are happening in the same year. Event 1 has durantion of two hours, when its happens the duration period is uninterrupted. Event 2 has ...
0
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2answers
23 views

Conditional Expectation: Sum inside or outside?

Let $X,Y$ be some discrete random variables with $Y$ taking values in $\mathbb{N}$ and consider $\mathbb{E}[X]$. Since it is sometimes easier to consider the expectation conditioned on a certain ...
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0answers
18 views

Interpretation of integral as ratio of joint and conditional densities?

A common exercise in Bayesian statistics is specifying a prior $p(\theta)$ on some parameter $\theta$. We then observe a collection of data $D=(X_1,\dots,X_N)$, the distribution of which is ...
0
votes
1answer
11 views

Find the required Chi-square score for an arbitrarily low p-value (2 degrees of freedom)

I'm trying to use the Chi-Square test to find the significance of data that suffers from the multiple testing problem. Because I have this multiple testing problem, the required p-value to view a test ...
0
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0answers
8 views

RBF transformation on a Normally Distributed Random Variable

I have a random vector $\mathbf{X} \sim \mathcal{N}(\mathbf{m,\Sigma})$ which is transformed by a Gaussian Radial Basis Function into the random variable $\mathbf{Y} = K(\mathbf X)$ where $K = ...
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votes
1answer
36 views

Basic Probability using combinations [on hold]

(a) A committee of 5 people is to be chosen from a group of 10 (6 men and 4 women) (i) How many committees of 5 members can be chosen from 10 people? (ii) How many of the committees from (a) will ...
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votes
1answer
16 views

Uniform distribution and real values [on hold]

If the random variable $k$ is uniformly distributed in $(0,5)$, What is the probability that the roots of the equation $4x^2+4xk + k + 2 = 0$ are real?
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2answers
49 views

If a professor has 7 students and they have to at least do 2 assignments each…

The professor has $7$ students. Each student has to do at least $2$ projects. There are $3$ projects: $A, B,$ and $C$. Project $A$ has been assigned $4$ times. $B$ has been assigned $5$ times. $C$ has ...
-1
votes
1answer
32 views

Bayes' theorem with multiple variables

On the page: https://en.wikipedia.org/wiki/Bayesian_inference#Formal_description_of_Bayesian_inference there is the result: $$p(\theta \mid \mathbf{X},\alpha) = \frac{p(\mathbf{X} \mid \theta) ...
0
votes
1answer
17 views

Slow convergence simulating log-normal sample from the normal

I am trying to simulate a log-normal random variable $Y$ with mean $m = \mathbb{E}[Y] = 0.001$ and standard deviation $s = 0.094$ by simulating a normal sample instead, and then exponentiating it. ...
0
votes
1answer
19 views

Finding a moment generating function

I want to find $M(t)$ of $$f(x)= \begin{cases} e^{(-x-1)} & \text{for } x > -1 \\ 0 & \text{otherwise} \end{cases}$$ $e^{(-x-1)}$ I tried to do $$\int_{-1}^{āˆž {}} e^{tx} ...
0
votes
2answers
44 views

Chance of failure of a machine in a year - Probability ?(Interview Question)

A machine has 3 components say A,B,C and at any given day chance of failure of any of them is 1%. The machine doesn't work if any of the component fails. So the machine doesn't work if either 1 / 2 / ...
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1answer
26 views

Conditional Expectation for IIDs [on hold]

What is E[X|X+Y=1] Given X and Y are independent and identically distributed.
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1answer
20 views

Probability of a user references in a network [on hold]

I am trying to figure out no of possible referrals of a user in a network. Where the size of a network is not fixed but we can set an assumption of 1000 persons. Edit: A user knows few users in a ...
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votes
2answers
89 views

What do you call this thing in probability theory? [on hold]

I have studied it before but I forgot the name. It is like when the possiblity of something happens is so small, but you created the experience so so many times, then the probability of that thing to ...
3
votes
1answer
17 views

CDF of the difference of two Gaussian mixtures

I have two Gaussian mixtures, $X_D$ and $X_{\overline{D}}$: $$ f(X_D) = \sum_{c=1}^m f(X_D\mid C=c)P(C=c) = \sum_{c=1}^m \phi(x-\mu-g(c))P(C=c), $$ $$ f(X_\overline{D}) = \sum_{c=1}^m ...
2
votes
1answer
78 views

Weird Induction…?

I was watching this video earlier and I couldnt figure out why the following step was possible. This is the original problem: $\sum_{i = 0}^{n} \binom{n + i}{i} = \binom{2n + 1}{n + 1}$ At one ...
0
votes
0answers
43 views

probability problem in a die game

I'm stuck with the following question: A, B and C are playing a game. At each turn, everyone tosses a fair die and the one with the largest number takes one dollar from the one with the least ...
0
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1answer
24 views

Probability the range is disjoint

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is ...
2
votes
1answer
45 views

Sum of squares of terms of a binomial expansion

I have a coin that show heads with a probability $p$. I toss it $N$ times and count the number of heads. I repeat the experiment once more. What's the probability that I get the same number of heads ...
2
votes
0answers
16 views

Tail field versus germ field of Brownian motion

Continuing my foray into Brownian motion (apologies for the bombardment...), I'm trying to verify the details of a proof of Durrett of the following 0-1 property of the tail $\sigma$-algebra of ...
1
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3answers
77 views

Why does $n \choose r$ where $r = 1,n$ track $2^n$?

I bashed together a clunky ruby script to find the sum total of $n \choose r$ where $r = 1,n$ I wanted to determine how many lines of output I could expect from a script that produces all possible ...
0
votes
1answer
42 views

Binomial Distribution Confidence Interval

Normally when I see confidence intervals it is in attempt to estimate a population parameter (probably poor wording). What I am trying to do is form a confidence interval for some theoretical values, ...
2
votes
1answer
29 views

Ask for a question about independence of random variable from an event

Consider two independent tosses of a fair coin. Let random variable X take the value 0 if the first toss is a head and take the value 1 if the first toss is a tail. Let A be the event that the number ...
3
votes
0answers
51 views

Proof that the sum does not depend on enumeration …

I've come across the following basic lemma in a basic probability book, and I can't seem to understand why the provided argument is enough to prove it. Lemma: Let $I$ be a countably infinite set and ...
2
votes
0answers
16 views

Central limit theorem in multidimension with unknown covariance

Let $X_1,\dots,X_n$ be samples from a distribution on $\mathbb{R}^d$ that has a finite second moment. If $d=1$, $\bar{X}_n=1/n\sum_{i=1}^nX_i$ and $S_n=1/(nāˆ’1)\sum_{i=1}^n(X_iāˆ’\bar{X}_n)^2$ then ...
0
votes
0answers
10 views

Formula for running-time complexity

I'm regarding a stochastic process $(X_t)$of which the mean starts at $O(n)$ and is reduced by the factor $(1-r)$ in each step with $r = \Omega (1/n^9)$, so $$E(X_{t+1}) \leq E(X_t) (1-r) .$$ Now it ...
1
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1answer
35 views

How is Poisson Distribution simply discerned? How is it related to the Binomial distribution?

There is this question which I thought I had understood, until taking a look at the answers: Let a floor tile be composed of different four tiles: a black one of size $1\times1$, a red $3\times 3$ ...
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votes
1answer
30 views

Does these inequalities hold in General for probability distribution? [on hold]

Let $Q(y)$ be a probability density of $y \in [-1,1]$. Then for $t> 0$, the inequalities are $\displaystyle \int_{0 \leq y <t} y^2 Q(y) \, dy \leq t^2 \int_{0 \leq y <t} Q(y) \, dy $. ...
2
votes
3answers
88 views

High computation in probability

Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at ...
2
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0answers
27 views

Expected number of times a set of 10 integers (selected from 1-100) is selected before all 100 are seen

Suppose I have a set of 100 integers. I randomly choose 10 of those, make a note of which ones I selected, and repeat the process. What is the expected number of times this process must be repeated ...
1
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0answers
69 views

Optimal allocation in network

Given a network (N,g). We want to analyse specializaton matters. Nodes are individuals, and they can product goods and services just like in our usual economy. Individuals can be consumers too. This ...
0
votes
1answer
17 views

computing weight from distance metric

I have a distance between two points in meters. I want to convert this distance into weight such that as distance increases the weight decreases. What are some good weighting function that can ...
2
votes
1answer
19 views

Probability in knockout games.

Suppose in a knockout tournament 32 players p1 , p2 .....p32 participate. In each round players are divided into pairs at random and winner goes to the next round. If p5 reaches semifinal what is ...
0
votes
0answers
17 views

Calculating Variance of payment in patterns of balls.

We have five different bags labeled from 1 to 5 and several colored balls. There are 9 different possible colors. We know how many balls of each color there are in each bag. We have a grid of 5x3 ...
1
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0answers
35 views

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) ...
1
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0answers
16 views

Distribution of sample skewness and kurtosis

I am working on my thesis right now and I'm almost done with it, but just on the last step I encountered some problems with a proof. I have an independent sample $X_{1}, ..., X_{n}$ that follows the ...
3
votes
6answers
55 views

Finding $P(X < Y)$ where $X$ and $Y$ are independent uniform random variables

Suppose $X$ and $Y$ are two independent uniform variables in the intervals $(0,2)$ and $(1,3)$ respectively. I need to find $P(X < Y)$. I've tried in this way: $$ \begin{eqnarray} P(X < Y) ...
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votes
1answer
40 views

Largest value of an expected value? [on hold]

I have $m$ balls and $n$ bins, and I want to find the expected number of non-empty bins. In order to make the problem easier, I decided to find the expected value of empty bins instead, but now im a ...
1
vote
0answers
24 views

Does convergence in probability preserve the weak inequality?

Suppose I have two sequences of random variables $\{x_n\}$ and $\{y_n\}$ such that $x_n\leq y_n$ and $\text{plim}\;x_n=L_x$ and $\text{plim}\;y_n=L_y$, can I say $L_x\leq L_y$ (almost surely)? Does ...
0
votes
2answers
33 views

A question on probability of choosing coins

Six identical-looking coins are in a box, of which five are unbiased, while the sixth comes up heads with probability $3 \over 4$ and tails with probability $1 \over 4$. Three coins are chosen from ...
1
vote
1answer
17 views

How do I use interpolation with the Z table?

My textbook has an example of interpolation, but I am not sure how the book did it since it doesn't explain it. It says if we want $P(Z < 1.246)$ we must use interpolation and the steps given are: ...
3
votes
2answers
81 views

Given a variable $X$ with a PDF, what is the PDF of $\sqrt{X}$

I feel this is simple and I'm overlooking something really basic. Let's say a have a variable $x$ which obeys the exponential distribution. So if collect 100000 occurrences of $x$ and plot its ...