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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3 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 = ...
-1
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1answer
30 views

Basic Probability using combinations

(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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1answer
15 views

Uniform distribution and real values

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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1answer
32 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 ...
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2answers
27 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) ...
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0answers
10 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. ...
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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} ...
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2answers
42 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
24 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
17 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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2answers
84 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 ...
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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 ...
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0answers
37 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 ...
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2answers
17 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
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1answer
42 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
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0answers
14 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 ...
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3answers
76 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 ...
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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
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1answer
27 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
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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
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0answers
14 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 ...
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0answers
8 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 ...
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1answer
34 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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1answer
21 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 $. ...
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votes
4answers
71 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 ...
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0answers
67 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 ...
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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
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1answer
18 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 ...
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0answers
15 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 ...
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0answers
33 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) ...
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0answers
15 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 ...
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6answers
52 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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1answer
39 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 ...
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0answers
23 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 ...
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2answers
32 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
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1answer
16 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
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2answers
78 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 ...
3
votes
1answer
25 views

Median of waiting time for $k$-th ace from bridge cards

I can't figure out how to get median of a waiting time from the exercise 36 from W. Feller's book An Introduction to Probability Theory and Its Applications Vol.1 (bold in the quote): ...
3
votes
1answer
24 views

Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi _{2n}^2$.

The Statement of the Problem: Suppose that $X_1,\ldots, X_n$ is a random sample from the $U(0,1)$ distribution and $$ Y_i = -2\log X_i. $$ Show that $Y = \sum_{i=1}^n Y_i$ is distributed as $\chi ...
0
votes
1answer
22 views

Probability involving a moment generating function

Suppose that X1 and X2 are independent and identically distributed discrete random variables. The moment generating function of X1 + X2 is: M(t)= 0.01e^(-2t) + 0.15e^(-t) +0.5925 + 0.225e^(t) + ...
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1answer
25 views

How to find median from a probability distribution?

Having trouble on something that should be really, really easy. I need to find the median of the following probability distribution...but according to the website I linked below...I'm doing it ...
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1answer
29 views

Lottery probability with payout system

Assume we have a lottery which has following payouts 1,2,5,6,9,10,16. The organizer expects 4% profit from the lottery. I wrote ...
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1answer
30 views

An application of Jensen's Inequality for dependent random variables

Consider dependent and positive valued random variables $A,B$ and $X$. I want to prove that \begin{equation} E[X^2 A] E[B] \ge E[X A] E[X B]. \end{equation} If $A$ and $B$ were scalars, above would ...
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2answers
22 views

Show that the conditional pmf of $X_i$, given $T = t$, is $Binomial(t, \lambda_i/\lambda).$

The Statement of the Problem: The random variables $X_1, ..., X_n$ are independent and $X_i \sim Poisson(\lambda _i), i = 1, ..., n$. Set $$ T = \sum_{i=1}^n X_i \qquad \text{and} \qquad \lambda = ...
2
votes
2answers
87 views

Sum of remainders of $2^n$

Hints Only Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by $1000$. Let $S$ be the sum of all elements in $R$. Find the ...
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0answers
21 views

What is the PMF of the Hamming weight of a multinomial random variable?

Assume that $X$ is a random variable following a multinomial distribution of parameters $n$ (number of trials) and $p=(p_1,\dots,p_k)$ (event probabilities). Hence, ...
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votes
1answer
29 views

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board

Given 5 colors to choose from, how many ways can we color the four unit squares of a $2\times 2$ board, given that two colorings are considered the same if one is a rotation of the other?
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3answers
29 views

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many

Given a 50 card deck with cards numbered from 1 through 10 in each of 5 suits, how many 5 card hands are there that include exactly one pair of two cards that have the same numeric value?